Abstract The paper is devoted to the study of shock formation of the 3-dimensional quasilinear wave equation
− (1 + 3G ′′(0)(∂tφ)2)∂2t φ + �φ = 0, (�)
where G ′′(0) is a non-zero constant. We will exhibit a family of smooth initialdata and show that the foliation of the incoming characteristic hypersurfacescollapses. Similar to 1-dimensional conservational laws, we refer this specifictype breakdown of smooth solutions as shock formation. Since (�) satisfiesthe classical null condition, it admits global smooth solutions for small data.Therefore, we will work with large data (in energy norm). Moreover, no sym-metry condition is imposed on the initial datum. We emphasize the geometricperspectives of shock formation in the proof. More specifically, the key ideais to study the interplay between the following two objects: (1) the energyestimates of the linearized equations of (�); (2) the differential geometry ofthe Lorentzian metric g = − 1
study of the null hypersurfaces of g. The techniques in the proof are inspiredby the work (Christodoulou in The Formation of Shocks in 3-DimensionalFluids. Monographs in Mathematics, European Mathematical Society, 2007)in which the formation of shocks for 3-dimensional relativistic compressibleEuler equations with small initial data is established. We also use the shortpulse method which is introduced in the study of formation of black holes ingeneral relativity in Christodoulou (The Formation of Black Holes in Gen-eral Relativity. Monographs in Mathematics, EuropeanMathematical Society,2009) and generalized in Klainerman and Rodnianski (ActaMath 208(2):211–333, 2012).
1 Introduction
This paper is devoted to the study of the following quasilinear wave equation
− (1 + 3G ′′(0)(∂tφ)2)∂2t φ + �φ = 0, (1.1)
where G ′′(0) is a nonzero constant and φ ∈ C∞(Rt × R3x ;R) is a smooth
solution.We propose a geometric mechanism for shock formation, i.e. how thesmoothness ofφ breaks down.We remark that ifG ′′(0) = 0, the equation is lin-ear so that no shock is expected.Wewill see that (�) can be regarded as the sim-plest quasilinear wave equations that can be derived from the least action prin-ciple. The equation can also be regarded as a model equation for the nonlinearversion ofMaxwell equations in nonlinear electromagnetic theory, inwhich theshocks can be observed experimentally. The shock formation in nonlinear elec-tromagnetic theory will be the subject of a forthcoming paper by the authors.
The breakdown mechanism is a central object in the theory of quasilinearhyperbolic equations. We give a brief account on the results related to thecurrent work. In [2], Alinhac proved a conjecture of Hörmander concerningupper bounds of the lifespan for the solutions of −∂2t φ + �φ = ∂tφ ∂
2t φ on
R2+1. This equationwas first introduced by John (see the survey paper [11] and
the references therein). He [10] studied the rotationally symmetric cases andobtained upper bounds for the lifespan of the solutions. In [2,3], without anysymmetry assumptions, Alinhac not only shows the solution blows up but alsogives a very precise description of the solution near the blow-up point. Despitethe slight different forms of the equations, Alinhac’s results are fundamentallydifferent from the currentwork in the following aspects: (1)He dealswith smalldata problem. We will (and have to) deal with large data problem. (2) He usesNash–Mosermethod to recover the loss of derivatives. Based on the variationalnature of (�), we can close the energy estimates with finite many derivatives.(3) Just as Alihnac’s work, we can give a detailed account on the behaviorsof the solutions near blow-up points. Moreover, we show that the singularities
123
On the formation of shocks for quasilinear wave equations
formed are indeed shocks. (4) We can give a pure geometric interpretation ofthe shock formation (in terms of certain curvature tensors) and show that theblow-up behavior indeed can be read off from the initial data directly.
A major breakthrough in understanding the shock formation for the Eulerequations has beenmade by Christodoulou in his monograph [5]. He considersthe relativistic Euler equations for a perfect irrotational fluid with an arbitraryequation of state. Provided certain smallness assumptions on the initial data, heobtained a complete picture of shock formation in three dimensions. A similarresult for classical Euler’s equations has also been obtained by Christodoulouand Miao in [8]. The approaches are based on differential geometric methodsoriginally introduced by Christodoulou and Klainerman in their monumentalproof [7] of the nonlinear stability of theMinkowski spacetime in general rela-tivity. Most recently, based on similar ideas, Holzegel, Klainerman, Speck andWong have obtained remarkable results in understanding the stablemechanismfor shock formation for certain types of quasilinear wave equations with smalldata in three dimensions, see their overview paper [9] and Speck’s detailedproof [14]. We remark that one of the key ideas in [5,8] is to explore the vari-ational structure of Euler’s equations. This idea also plays a key rôle in thecurrent work. We emphasize that [5,8] obtained sharp lower and upper boundsfor the lifespan of smooth solutions associated to the given data without anysymmetry conditions. Prior to [5,8], most of works on shock waves in fluidare limited to the simplified case of with spherical symmetry assumptions, i.e.essentially the one space dimension case. As an example, we mention [1] ofAlihnac which studies the singularity formation for the compressible Eulerequations on R
2 with rotational symmetry.All the aforementioned works have the common feature that the initial data
are assumed to be small. However, since the nonlinearity in (�) is cubic. By theclassical result of Klainerman [12], for small smooth initial data, the solutionsof (�) are globally regular. In particular, we do not expect shock formation.Wewill use a special family of large data, so called short pulse data, in the currentwork. It was firstly introduced by Christodoulou in a milestone work [6] inunderstanding the formation of black holes in general relativity. By identifyingan open set of initial data without any symmetry assumptions (the short pulseansatz!), he shows that a trapped surface can form, even in vacuum space-time, from completely dispersed initial configurations and by means of thefocusing effect of gravitational waves. Although the data are no longer closetoMinkowski data, in otherwords, the data are no longer small, he is still able toprove a long time existence result for these data. This establishes the first resulton the long time dynamics in general relativity and paves the way for manynew developments on dynamical problems related to black holes. Shortly afterChristodoulou’s work, Klainerman and Rodnianski extends and significantlysimplifies Christodoulou’s work, see [13]. From a pure PDE perspective, the
123
S. Miao, P. Yu
data appeared in the above works are carefully chosen large profiles whichcan be preserved by the Einstein equations along the evolution. The data inthe current work are inspired by these ideas, in particular the idea in [6]. Theinitial profiles are designed in such a way that the shape of the data will bepreserved along the evolution of (�).
As a summary, the data used in the paper are motivated by Christodoulou’swork [6] and Klainerman-Rodnianski [13] on the formation of black holes ingeneral relativity. The ideas of the proof aremotivated byChristodoulou’swork[5] and Christodoulou–Miao [8]. We have to overcome all the technical diffi-culties in the works mentioned above, in particular those in Christodoulou’sworks. At the same time, we would like to present a clearer geometric pictureof the underlying shock formation mechanism.
1.1 The heuristics for shock formation
We rewrite (�) in the so called geometric form:
− 1
c2∂2t φ + �φ = 0 · · · · · · (�g),
where c = (1+3G ′′(0)(∂tφ)2)− 1
2 . Recall that, if c was a constant, (�g)woulddescribe the propagation of light in Minkowski space and c was the speed oflight. In the current situation, we still regard c (which is not a constant) as thespeed of light. But the speed of light depends on the position (t, x) in spacetimeand the solution φ. This is of course the quasilinear nature of the equation. Wenow briefly review on the basics of shock formation for the inviscid Burgers’equation. The idea is to get a heuristic argument for the main equation (�) andto motivate the main theorem.
The inviscid Burgers’ equation can be written as
∂t u + u∂xu = 0 · · · · · · (∗).
We assume that u ∈ C∞(Rt × Rx ;R) is a smooth solution. Given smoothinitial datum u(0, x)(non-zero everywhere for simplicity), (∗) can be solvedby the method of characteristics. A characteristic is a curve in R2
t,x defined bythe solution u. In the case of Burgers’ equation, a characteristic is a straightline and it is determined by the initial datum u(0, x) as follows: it is the uniqueline passing through (0, x) with slope 1
u(0,x) . The method of characteristicssays that u is constant along each of the characteristics.
To make connections to the geometric form (�g) of the main equation (�),we also propose a geometric form of the Burgers’ equation (we assume thatu �= 0 to make the following computation legitimate):
123
On the formation of shocks for quasilinear wave equations
1
c∂t u + ∂xu = 0 · · · · · · (∗g),
where c = u. Then c is the speed of the characteristics and it depends on thesolution u.
We now consider two specific characteristics passing through x1 and x2(x1 < x2). Ifwe choose datum in such away that u(0, x1) > u(0, x2) > 0, boththe characteristics travel towards the right. Moreover, the characteristic on theleft (noted asC1) travels with speed c1 = u(0, x1) and the characteristic on theright (noted asC2) travelswith speed c2 = u(0, x2). SinceC1 travels faster thanC2, C1 will eventually catch up with C2. The collision of two characteristicscauses the breakdown on the smoothness of the solution. In summary, we havea geometric perspective on shock formation: a “faster” characteristic catchesup a “slower” one so that it causes a collapse of characteristics.
The above discussion can also be read off easily from the following picture:
In reality, in stead of showing that characteristics collapse (which is onthe heuristic level), we show that |∂xu| blows up. In stead of being naïvely aderivative, |∂xu| have an important geometric interpretation. Recall that thelevel sets of u are exactly the characteristics and the (t, x)-plane is foliated bythe characteristics (see the above picture). Therefore, |∂xu| is the density of thefoliation by the characteristics. As a consequence, we can regard the shockformation as the following geometric picture: the foliation of characteristicsbecomes infinitely dense.
We also recall a standard way to prove the blow-up of |∂xu|. The remarkablefeature of this standard proof is that in three dimensions similar phenomenonhappens for themain equation (�). Let L = ∂t+u∂x be the generator vectorfieldof the characteristics (for (�), the corresponding vectorfield are generators ofnull geodesics on the characteristic hypersurfaces). Therefore, by taking ∂x
123
S. Miao, P. Yu
derivatives, we obtain
L∂xu + (∂xu)2 = 0.
This is a Riccati equation for ∂xu and the blow-up theory for ∂xu is standard.However, we would like to understand the blow-up in another way (which isintimately tied to the shock formation for (�)). We define the inverse densityfunction μ = −(∂xu)−1, therefore, along each characteristic curve, μ satisfiesthe following equation:
Lμ(t, x) = −1,
i.e. Lμ is constant along each characteristic so that it is determined by itsinitial value. Therefore, μ will eventually become 0 which implies that thefoliation becomes infinitely dense (For (�), we will also define an inverse den-sity function μ for the foliation of characteristic hypersurfaces and show thatLμ(t, x) is almost a constant along each generating geodesic of the character-istic hypersurfaces).
We return to the main equation in the geometric form − 1c2∂2t φ +
�φ = 0 with c = (1 + 3G ′′(0)(∂tφ)2)−12 . We prescribe initial data
(φ(−2, ·), ∂tφ(−2, ·)) on the time slice �−2 defined by t = −2. We useSr to denote the sphere of radius r centered at the origin on�−2 and use B2 todenote the ball of radius 2 with boundary S2. Therefore, the region enclosedby S2 and S2+δ (where δ is a small positive number) is foliated by the Sr ’s for2 ≤ r ≤ 2 + δ. The following picture may help to illustrate the process.
For each leaf Sr in the foliation, there is a unique incoming characteris-tic hypersurface, which will be defined more precisely in the next section,emanated from Sr . In the picture, we use a blue surface to denote it. The
123
On the formation of shocks for quasilinear wave equations
incoming characteristic hypersurfaces emanated from S2 and S2+δ are drawnin black and red respectively.
(1) Data inside B2. We take trivial initial data φ(−2, x) ≡ 0 and∂tφ(−2, x) ≡ 0 inside B2.
In view of the Huygens’ principle, in the backward solid light cone withbottom B2 (colored in black in the picture), the solution φ is identicallyzero. In particular, for the incoming characteristic hypersurface, which isthe cone in black in the picture emanated from the leaf S2, since c =(1 + 3G ′′(0)(∂tφ)2)−
12 , the incoming speed of this hypersurface, which is
noted as c1, can be computed as
c1 = 1.
(2) Data on the annulus region between S2 and S2+δ. This is the regionbetween the black circle and the red circle in the picture. We require that the
size of ∂tφ is approximately δ12 on the outermost circle (which is red in the
picture) S2+δ, i.e. |∂tφ| ∼ δ12 at S2+δ .
By Taylor expansion, we can compute the speed c2 of the outermost incom-ing characteristics hypersurface emanated from S2+δ (which is red in thepicture) as follows
c2 = 1 − 3
2G ′′(0)δ + O(δ2).
We are now in a situation that resembles the Burgers’ picture. The initialdistance between the inner most characteristic hypersurface (which is black inthe picture) and the outer most characteristic hypersurface (which is red in thepicture) is δ. Both hypersurfaces travel towards the center. The difference ofthe speeds of two characteristic hypersurfaces is c1 − c2 ∼ δ. We also expectthe “faster”(outer) characteristic hypersurface catching up the “slower”(inner)one. This catching up process needs approximately distance
speed = δc1−c2
∼ 1amount of time. We also regard the collision of characteristic hypersurfacesas shock formation, we hope that shocks form around t = −1.
We would like to point out a serious gap in the above heuristic argument.There is one assumption which seems to be very unreasonable: by the choiceof the data, we can make sure that the speed c2 of the outer most incomingcharacteristic hypersurfaces is of size 1 − 3
2G′′(0)δ + O(δ2), but there is no
clear reason that we should believe the speed c2 remaining the same later on.Therefore, the difference of the two speeds c1 − c2 may vary a lot so that theouter most characteristic hypersurface never catch up with the inner one.
The whole point of the paper is to identify a set of initial data so that theprofile of the data propagates, i.e. the profile remains almost unchanged. In
123
S. Miao, P. Yu
particular, we can prove that the speeds of the characteristic hypersurfacesremain almost unchanged for later time. Another way to understand this isthrough energy estimates: we can find a specific set of data so that we canobtain a priori energy estimates. Once we showed that the energy (and itshigher order analogue) is almost conserved, we can use Sobolev inequalityto show that ∂tφ is almost conserved pointwisely along the generators of theincoming characteristic hypersurfaces. According to the formula of c, this alsoimplies the speeds are almost conserved.
Finally, we point out that, as in the Burgers’ equation case, instead of show-ing that characteristic hypersurfaces meet, we show that the inverse density μof the foliation by the characteristic hypersurfaces becomes 0, i.e. the foliationturns to being infinitely dense. Similarly, this can be done by showing thatLμ(t, x) is almost a constant along each generating geodesic of the character-istic hypersurfaces.
1.2 The main result
With motivations from the previous subsection, we are ready to state the mainresult of the paper. Let t be the time function in Minkowski spacetime. We use�t to denote the level sets of t and it is a copy of R3 for each t . We fix r0 = 2in this paper.1 We also use �δ−r0 to denote the following δ-thin annulus:
�δ−r0 := {x ∈ �−r0
∣∣r0 ≤ r(x) ≤ r0 + δ}, (1.2)
where δ is any given small positive constant.
We recall that the wave speed c is defined as c = (1 + 3G ′′(0)(∂tφ)2)−12 .
Let L = ∂t − c∂r and L = ∂t + c∂r . We first introduce a pair of functions(φ1(s, θ), φ2(s, θ)) ∈ C∞((0, 1] × S
2) and we will call it the seed data.The seed data (φ1(s, θ), φ2(s, θ)) can be freely prescribed and once it is
given once forever. In particular, the choice of the seed data is independent ofthe small parameter δ.
Lemma 1.1 Given seed data (φ1, φ2), there exists a δ′ > 0 depending onlyon the seed data, for all δ < δ′, we can construct another function φ0 ∈C∞((0, 1] × S
2) satisfying the following two properties:1. For all k ∈ Z≥0, the Ck-norm of φ0 are bounded by a function in the
Ck-norms of φ1 and φ2;2. If we pose initial data for (�) on �−2 in the following way:1 In a future work, we will consider a more general case for which the data is prescribed atpast infinity. Therefore, we have to let r0 go to ∞ and the dependence of the estimates (of thecurrent work) on r0 will be crucial.
123
On the formation of shocks for quasilinear wave equations
For all x ∈ �−2 with r(x) ≤ 2, we require (φ(−2, x), ∂tφ(−2, x)) =(0, 0); For 2 ≤ r(x) ≤ 2 + δ, we require that
φ(−2, x) = δ3/2φ0
(r − 2
δ, θ
), (∂tφ)(−2, x) = δ1/2φ1
(r − 2
δ, θ
).
Then we have
‖Lφ‖L∞(�δ−r0
) � δ3/2, ‖L2φ‖L∞(�δ−r0
) � δ3/2. (1.3)
We remark that the condition (1.3) has a clear physical meaning: since Lare incoming directions, the waves are initially set to be incoming and theoutgoing radiation is very little (controlled by δ).
Definition 1.2 The Cauchy initial data of (�) constructed in the lemma (sat-isfying the two properties) are called no-outgoing-radiation short pulse data.
Before we state the main theorem, we prove the lemma hence show theexistence of no-outgoing-radiation short pulse data.
Proof We recall that c = (1 + 3G ′′(0)(∂tφ)2)−12 , L = ∂t − c∂r and L =
∂t + c∂r .We first take an arbitrary choice of φ1 and fix this function. Therefore, ∂tφ
is given by the formula (∂tφ)(−2, x) = δ1/2φ1(r−2δ, θ). In particular, all the
spatial derivatives of ∂tφ and c (determined completely by ∂tφ) are prescribedon �δ−2.
Therefore, by definition, we have
L2φ = ∂2t φ − ∂t c ∂rφ + c∂r c ∂rφ + c2∂2r φ − 2c∂r (∂tφ).
According to the definition of c, we have ∂t c = −3G ′′(0)c3∂tφ ∂2t φ. Thus, wehave
We claim that we can choose φ0, which may depend on the choice of φ1 but
is independent of δ, in such a way that |L2φ| � δ32 .
To see this, we first observe that since φ1 is given and δ is small, we have|c|+ |∂r c| � 1. Indeed, ∂r c = −3c3G ′′(0)φ1∂sφ1 so the bound on ∂r c is clear.We make the following ansatz for φ0:
|∂sφ0| + |∂2θ φ0| ≤ C, (1.5)
where the constant C may only depend on φ1 but not on δ.
By the ansatz (1.5) and by looking at the expansions in δ12 , one can ignore
all the terms equal to or higher than δ32 . Therefore, to show |L2φ| � δ
32 , it
suffices to consider
(2 − 3G ′′(0)c3φ1 ∂sφ0 δ
)(δ− 1
2 c2∂2s φ0)+ 2c2
r∂sφ0 δ
12
+ c∂r c ∂sφ0 · δ 12 − 2c∂sφ1δ
− 12 = O(δ
32 ),
or equivalently
(2 − 3G ′′(0)c3φ1 ∂sφ0 δ
)(c2∂2s φ0
)+ 2c2
r∂sφ0 δ
+ c∂r c ∂sφ0 δ − 2c∂sφ1 = O(δ2).
Since (2 − 3G ′′(0)c3φ1 ∂sφ0 δ)−1 = 12 + 3
4G′′(0)c3φ1 ∂sφ0 δ + O(δ2), by
multiplying both sides of the above identity by (2− 3G ′′(0)c3φ1 ∂sφ0 δ)−1, itsuffices to consider
c2∂2s φ0 + c2
r∂sφ0 δ + 1
2c∂r c ∂sφ0 δ
−(1 + 3
2G ′′(0)c3φ1 ∂sφ0 δ
)c∂sφ1 = O(δ2).
123
On the formation of shocks for quasilinear wave equations
Since c ∼ 1, we finally have
∂2s φ0 +(δ
r+ δ
2c∂r c − 3δ
2G ′′(0)c2φ1∂sφ1
)∂sφ0 − c−1∂sφ1 = O(δ2).
To solve for φ0, for s ∈ [0, 1), we consider the following family of (para-metrized by a compact set of parameters θ ∈ S
2 and the parameter δ) linearordinary differential equation:
∂2s φ0 +(δ
r+ δ
2c∂r c − 3δ
2G ′′(0)c2φ1∂sφ1
)∂sφ0 − c−1∂sφ1 = δ2φ2,
φ0(0, θ) = 0, ∂sφ0(0, θ) = 0.
Since the Ck-norms of the solution depends smoothly on the coefficients andthe parameter θ, δ, all Ck-norms of φ0 are of order O(1) and indeed are deter-mined by the solution of
∂2s φ0 − c−1∂sφ1 = 0,
φ0(0, θ) = 0, ∂sφ0(0, θ) = 0.
In particular, this shows that the ansatz (1.5) holds ifwe chooseC appropriatelylarge in (1.5) and δ sufficiently small. Therefore the above construction showsthat
|L2φ| � δ32 .
We claim that, by the above choice of initial data, on �δ−2, we automaticallyhave
|Lφ| � δ32 .
Indeed, by replacing ∂t = L + c∂r in the main equation, we obtain
∂r Lφ = 1
2c
(−L2φ − Lc ∂rφ + 2c2
r∂rφ + c2
r2�/S2φ
).
By the construction of the data, it is obvious that all the terms on the right
hand side are of size O(δ12 ). By integrating from 2 to r with r ∈ [2, 2 + δ)
and Lφ(2, θ) = 0, we have
|Lφ(r, θ)| ≤ δ · O(δ12 ) � δ
32 .
�
123
S. Miao, P. Yu
The main theorem of the paper is as follows:
Main Theorem For a given constant G ′′(0) �= 0, we consider
−(1 + 3G ′′(0)(∂tφ)2)∂2t φ + �φ = 0.
Let (φ1, φ2) be a pair of seed data and the initial data for the equation is takento be the no-outgoing-radiation initial data.
If the following condition on φ1 holds for at least one (r, θ) ∈ (0, 1] × S2:
G ′′(0) · ∂rφ1(r, θ) · φ1(r, θ) ≤ −1
6, (1.6)
then there exists a constant δ0 which depends only on the seed data (φ1, φ2),so that for all δ < δ0, shocks form for the corresponding solution φ beforet = −1, i.e. φ will no longer be smooth.
Remark 1.3 The choice ofφ1 in the proof of Lemma 1.1 is arbitrary. In particu-lar, this is consistentwith the condition (1.6) since φ1 can be freely prescribed.
Remark 1.4 1. We do not assume spherical symmetry on the initial data.Therefore, the theorem is in nature a higher dimensional result.
2. The proof can be applied to a large family of equations derived throughaction principles. We will discuss this point when we consider theLagrangian formulation of (�).
3. The condition (1.6) is only needed to create shocks. It is not necessary atall for the a priori energy estimates.
Remark 1.5 The smoothness of φ breaks down in the following sense:
1. The solution and its first derivative, i.e. φ and ∂φ, are always bounded.
Moreover, |∂tφ| � δ12 , therefore (�) is always of wave type.
2. The second derivative of the solution blows up. In fact, when oneapproaches the shocks, ∇∂tφ blows up. See Remark 4.7 for the proof.
1.3 Lagrangian formulation of the main equation and its relation tononlinear electromagnetic waves
We briefly discuss the derivation of the main equation (�). The linear waveequation inMinkowski spacetime (R3+1,mμν) can be derived by a variationalprinciple: we take the Lagrangian density L(φ) to be 1
2 (−(∂tφ)2 + |∇xφ|2)
and take the action functional L(φ) to be ∫R3+1 L(φ)dμm where dμm denotes
the volume form of the standard Minkowski metric mμν . The correspondingEuler–Lagrange equation is exactly the linear wave equation−∂2t φ+�φ = 0.
123
On the formation of shocks for quasilinear wave equations
We observe that the quadratic nature of the Lagrangian density result in thelinearity of the equation. This simple observation allows one to derive plenty ofnonlinear wave equations by changing the quadratic nature of the Lagrangiandensity. In particular, we will change the quadratic term in ∂tφ to a quarticterm, this will lead to a quasi-linear wave equation.
In fact,we consider a perturbation of theLagrangian density of linearwaves:
L(φ) = −1
2G((∂tφ)
2)+ 1
2|∇φ|2, (1.7)
where G = G(ρ) is a smooth function defined on R and ρ = |∂tφ|2. Thecorresponding Euler–Lagrange equation is
−∂t(G ′(ρ)∂tφ
)+ �φ = 0.
The function G(ρ) as a perturbation of G0(ρ) = ρ and therefore we canthink of the above equation as a perturbation of the linear wave equation. Forinstance, we can work with a real analytic function G(ρ) with G(0) = 0 andG ′(0) = 1. In particular, we can perturb G(ρ) = ρ in the simplest possibleway by adding a quadratic function so that G(ρ) = ρ + 1
2G′′(0)ρ2. In this
situation, we obtain precisely the main equation (�). It is in this sense that(�) can be regarded as the simplest quasi-linear wave equation derived fromaction principles.
The main equation (�) is also closely tight to electromagnetic waves ina nonlinear dielectric. The Maxwell equations in a homogeneous insulatorderived from a Lagrangian L which is a function of the electric field E andthe magnetic field B. The corresponding displacements D and H are definedthrough L by D = − ∂L
∂E and H = ∂L∂B respectively. In the case of an isotropic
dielectric, L is of the form
L = −1
2G(|E |2) + 1
2|B|2, (1.8)
hence H = B. The fields E and B are derived from the scalar potential φand the vector potential A according to E = −∇φ − ∂t A and B = ∇ × Arespectively. This is equivalent to the first pair of Maxwell equations:
∇ × E + ∂t B = 0, ∇ · B = 0.
The potentials are determined only up to a gauge transformation φ �→ φ−∂t fand A �→ A + d f , where f is an arbitrary smooth function. The second pairof Maxwell equations
∇ · D = 0, ∇ × H − ∂t D = 0
123
S. Miao, P. Yu
are the Euler–Lagrange equations, the first resulting from the variation of φand the second resulting from the variation of A. Fixing the gauge by settingφ = 0, we obtain a simplified model if we neglect the vector character of Areplacing it by a scalar function φ. Then the above equations for the fields interms of the potentials simplify to E = −∂tφ and B = ∇φ. The Lagrangian(1.8) becomes
L = −1
2G((∂tφ)
2) + 1
2|∇φ|2
which is exactly (1.7). Therefore, the main equation (�) provides a goodapproximation for shock formation in a natural physical model: the shockformation for nonlinear electromagnetic waves.
1.4 Main features of the proof
We now briefly sketch four main ingredients of the proof.1. The short pulse method. By rewriting (�) in the semilinear form �mφ =
−3G ′′(0)(∂tφ)2∂2t φ, we notice that the nonlinearity is cubic. Therefore theresult in [12] implies that small smooth initial data lead to global smoothsolutions since the classical null condition is satisfied. We are then forcedto consider large initial data. According to the choice of data in the Main
Theorem, they are supported in the annulus of width δ and with amplitude δ12
which looks like a pulse (the short pulse data). The energy associated to thedata is of size 1. On the technical level, although the short pulse data is nolonger small, we still have a small parameter δ coming into play. Therefore,most of the techniques for small data problems can also be applied here.
2. A Lorentzian geometry defined by the solutions. Since the Lagrangian,therefore (�) itself, is invariant under the time translation and the isometriesof R3, we can linearize the equation via the infinitesimal generators of thoseactions. The most important feature about the linearized equations is, theyare not just linear, they are linear wave equations with respect to a spe-cial Lorentzian metric defined by the solution. This reflects the Lagrangiannature of the equation (�): the metric comes from the second derivative of theLagrangian. In particular, the incoming null hypersurfaces with respect to thismetric correspond to the characteristic hypersurfaces of the solution. Recallthat the shock formation is the study of the collapsing of the characteristichypersurfaces, hence the differential geometry of the metric dictates shockformation.
Moreover, we study the energy estimates for the linearized linear waveequations. The energy estimates on one hand depend heavily on the under-lying geometry, e.g. the curvature, the fundamental forms of null foliations,
123
On the formation of shocks for quasilinear wave equations
the isoperimetric inequalities, etc.; on the other hand, the energy estimatesalso control the underlying geometry. Therefore, the study of the linearizedequations are more or less equivalent to the study of the underlying geometry.This leads to a natural bootstrap argument.
3. Coercivity of angular energy near shocks. We use the vector field methodto study the energy estimates for the linearized equations. Since we expectshock waves, the function μ, i.e., the inverse density of the characteristichypersurfaces,may turn to 0. Thiswill pose a fundamental difficulty for energyestimates (even for linear wave equations!). Roughly speaking, for a free waveψ , for all possible multiplier vectorfields, in the associated energy or fluxintegrals, the components for the rotational directions all look like
∫μ|∇/ψ |2.
But in the error integrals, some ∇/ψ components show up without a μ factor.In view of the fact that μ → 0 in the shock region, the above disparity in μshows that one can not control the error integrals by the energy or flux terms.
This difficulty is of course tied to the formation of shocks. The remarkablething is, it is also resolved by the formation of shocks. The idea is as follows:initially, the μ ∼ 1. If in the future, no shock forms, then the disparity of μsimply result in an universal constant in the estimates since μwill be boundedbelow and above. If shock forms eventually, then along the incoming directionμ decreases, i.e. Lμ < 0 where L is the generator of the incoming nullgeodesics. Although the error integrals contain many terms without factor μfor ∇/ψ , there is one term has a very special form: it looks like
∫∫Lμ|∇/ψ |2.
The sign of Lμ in the shock region shows amiraculous coercivity of the energyestimates. This term is just enough to control all the∇/ψ terms appearing in theerror terms. This is the major difference between the usual energy estimatesand the case where shocks form. The use of the sign Lμ is the key to the entireargument in the current work.
4. The descent scheme. The energy estimates on the top order terms maysuffer a loss of a factor in μ and this can be dangerous in the shock region.Indeed, some error integral looks like
∫ t−r0
μ−1 ∂μ∂t E(τ )dτ where E(τ ) for the
energy (it appears also on the lefthand side of the energy identity). If s∗ isthe time where shock forms, we can show that μ behaves like |t − s∗| nearshocks. Therefore, the presence of μ−1 cause a log loss in time. The descentscheme is designed to retrieve the loss. The idea is, rather than proving thetop order terms are bounded in energy, we prove that the energy may blowup with a specific rate in μ to some negative power. To illustrate the idea, wedo the following formal computations by assuming E(t) = supτ≤t μ
a E(τ ) isbounded for a large positive number a. The energy identity, which looks likeE(t) + · · · �
∫ t−r0
μ−1 ∂μ∂t E(τ )dτ + · · · , can be rewritten as
E(t) + · · · �(∫ t
−r0μ−(a+1) ∂μ
∂tdτ
)E(t) + · · · .
123
S. Miao, P. Yu
Since μ ∼ |t − s∗|, the term in the parenthesis gives a factor 1a which is small
(a is large!). Therefore, the righthand side can be absorbed by the lefthandside.
2 The optical geometry
2.1 Optical metrics and linearized equations
Weobserve thatmain equation (�) is invariant under the following symmetries:space translations, rotations and the time translation. Indeed, the LagrangianL(φ) is invariant under these symmetries, hence the Euler–Lagrange equa-tion must be invariant too. We use A to denote any possible choice from{∂t , ∂i , i j = xi∂ j − x j∂i } where i, j = 1, 2, 3 and i < j . These vectorfieldscorrespond to the infinitesimal generators of the symmetries of (�).
We linearize (�) according to A by the following procedure: We apply thesymmetry generated by A to a solution φ of (�) to obtain a family of solutions{φτ : τ ∈ R
∣∣φ0 = φ}. Therefore,− 1
c(φτ )2∂2t φτ +�φτ = 0 for τ ∈ R. We then
differentiate in τ and evaluate at τ = 0. We define the so called variationsψ as
ψ := Aφ = dφτdτ
|τ=0 (2.1)
By regarding φ as a fixed function, this procedure produce a linear equationfor ψ . We call it the linearized equation of (�) for the solution φ with respectto the symmetry A.
In the tangent space at each point in R3+1 where the solution φ is defined,
with (t, x1, x2, x3) being the standard rectangular coordinates in Minkowskispacetime. Since c depends on the solution φ, gμν also depends on the solutionφ. We also introduce a conformal metric gμν with the conformal factor = 1
c
gμν = · gμν = 1
cgμν. (2.3)
We refer gμν and gμν as the optical metric and the conformal optical metricrespectively.
123
On the formation of shocks for quasilinear wave equations
Lemma 2.1 The linearized equation of (�) for a solution φ with respect to Acan be written as
�gψ = 0, (2.4)
where �g is the wave operator with respect to g and ψ = Aφ.
There are two ways to derive the linearized equations.To derive (2.4), we can directly differentiate (�). We denote the Christoffel
symbols of g in the Cartesian coordinates by �γαβ . Let �
γ = gαβ�γαβ , then
�0 = −2c−2∂t c and the other �γ ’s vanish. Hence,
�gψ = gμν∂μ∂νψ − �γ ∂γ ψ
= c
⎡
⎢⎢⎢⎣
(− 1
c2∂2t ψ + �ψ
)
︸ ︷︷ ︸T1
− ∂ρ
(1
c2
)∂tρ · ∂tψ
︸ ︷︷ ︸T2
⎤
⎥⎥⎥⎦,
where ρ = (∂tφ)2. We use A to differentiate (�g). If A hits the factor − 1
c2, it
yields T2; since A (the symmetries!) commutes with ∂t and�, the other termsare precisely T1.
There is a more natural proof which is standard in Lagrangian field theory,e.g. see [4]. In fact, the linearized equation of (�) is the Euler–Lagrange equa-tion of the linearized Lagrangian density L(ψ) := 1
2d2
dτ 2
∣∣τ=0L(φ+τψ). Since
G(ρ) = ρ + 12G
′′(0)ρ2, we have
L(ψ)= − 1
2G ′(0)(∂tψ)2−3
2G ′′(0)(∂tφ)2(∂tψ)2 + 1
2|∇ψ |2= 1
2gμν∂μψ∂νψ.
Therefore, if D ⊂ R3+1 is a domain in which the solution φ is defined, the
action corresponding to L is L(ψ) = 12
∫D gμν∂μψ∂νψ dμm . We emphasize
that the volume form dμm is defined by the Minkowski metric mαβ . In viewof the definitions of gμν and gμν , the action L(ψ) can be written as
L(ψ) = 1
2
∫
Dgμν∂μψ∂νψ dμg.
At this stage, it is clear that the linearized equation is the free wave equationwith respect to �g.
123
S. Miao, P. Yu
2.2 Lorentzian geometry of the maximal development
2.2.1 The maximal development
We define a function u on �−2 as follows:
u := r − 2. (2.5)
The level sets of u in �−2 are denoted by S−2,u and they are round spheres ofradii u + 2. The annular region �δ−2 defined in (1.2) is foliated by S−2,u as
�δ−2 :=⋃
u∈[0,δ]S−2,u . (2.6)
Given an initial data set (φ, ∂tφ)∣∣t=−2 defined on B2+δ
−2 =⋃u∈[−2,δ] S−2,uto the main equation (�) (as we stated in the Main Theorem), we recall thenotion of the maximal development or maximal solution with respect to thegiven data.
By virtue of the local existence theorem (to (�) with smooth data), one canclaim the existence of a development of the given initial data set, namely, theexistence of
• a domain D in Minkowski spacetime, whose past boundary is B2+δ−2 ;
• a smooth solution φ to (�) defined on D with the given data on B2+δ−2
with following property: For any point p ∈ D, if an inextendible curveγ : [0, τ ) → D satisfies the property that1. γ (0) = p,2. For any τ ′ ∈ [0, τ ), the tangent vector γ ′(τ ′) is past-pointed and causal
(i.e., g(γ ′(τ ′), γ ′(τ ′)) ≤ 0) with respect to the optical metric gαβ at thepoint γ (τ ′),
then the curve γ must terminate at a point of B2+δ−2 .
123
On the formation of shocks for quasilinear wave equations
By the standard terminology of Lorentzian geometry, the above simply saysthat B2+δ
−2 is a Cauchy hypersurface of D.The local uniqueness theorem asserts that if (D1, φ1) and (D2, φ2) are two
developments of the same initial data sets, thenφ1 = φ2 inD1⋂D2. Therefore
the union of all developments of a given initial data set is itself a development.This is the so called maximal development and its corresponding domain isdenoted by W ∗. The corresponding solution is called the maximal solution.Sometimes we also identify the development as its corresponding domainwhen there is no confusion.
2.2.2 Geometric set-up
Given an initial data set, we consider a specific family of incoming null hyper-surfaces (with respect to the optical metric g) on the maximal developmentW ∗. Recall that u is defined on �−2 as r − 2. For any u ∈ [0, δ], we use Cuto denote the incoming null hypersurface emanated from the sphere S−2,u . Bydefinition, we have Cu ⊂ W ∗ and Cu
⋂�−2 = S−2,u .
We denote the subset of the maximal development of the given initial datafoliated by Cu with u ∈ [0, δ] by Wδ , i.e.,
Wδ =⋃
u∈[0,δ]Cu . (2.7)
Roughly speaking, our main estimates will be carried out only on Wδ . Thereason is as follows: since we assume that the data set is completely trivialfor u ≤ 0 on �−2, the uniqueness of smooth solutions for quasilinear waveequations implies that the spacetime in the interior of C0 is indeed determinedby the trivial solution. Modulo the spherical configurations, the situation canbe depicted as
123
S. Miao, P. Yu
The grey region is Wδ . In the light gray region, the solution φ ≡ 0 thanksto the construction of the initial data. The solution to (1.1) then vanishes up toinfinite order on C0, which is part of the boundary of Wδ . In particular, C0 isa flat cone in Minkowski spacetime (with respect to the Minkowski metric).
The dashed line denotes a incoming null hypersurface in the above picture.We extend the function u to Wδ by requiring that the hypersurfaces Cu areprecisely the level sets of the function u. Since Cu is null with respect to gαβ ,the function u is then a solution to the equation
(g−1)αβ∂αu∂βu = 0, (2.8)
where (g−1)αβ is the inverse of the metric gαβ . We call such a function u anoptical function.
With respect to the affine parameter, the future-directed tangent vectorfieldof a null geodesic on Cu is given by
L := −(g−1)αβ∂αu ∂β. (2.9)
However, for an apparent reason, which will be seen later, instead of usingL , we will work with a renormalized (by the time function t) vectorfield Ldefined through
L = μL, Lt = 1, (2.10)
i.e., L is the tangent vectorfield of null geodesics parametrized by t .The function μ can be computed as
1
μ= −(g−1)αβ∂αu∂β t.
We will see later on that the μ also has a very important geometric meaning:μ−1 is the density of the foliation
⋃u∈[0,δ] Cu .
123
On the formation of shocks for quasilinear wave equations
Given u ≤ δ, to consider the density of null-hypersurface-foliation on�t⋂
Wδ , we define
μum(t) = min
(
inf(u′,θ)∈[0,u]×S2
μ(t, u′, θ), 1)
. (2.11)
For u = δ, we define
s∗ = sup{t∣∣t ≥ −2 and μδ
m(t) > 0}.
From the PDE perspective, for the given initial data to (�) (as constructed inLemma 1.1), we also define
t∗ = sup{τ∣∣τ ≥ −2 such that the smooth solution exists for all
(t, u) ∈ [−2, τ ) × [0, δ] and θ ∈ S2}.
Finally, we define
s∗ = min{s∗,−1}, t∗ = min{t∗, s∗}. (2.12)
We remark that we will exhibit data in such a way that the solution breaksdown before t = −1. This is the reason we take −1 in the definition of s∗.
In the sequel, we will work in a further confined spacetime domain W ∗δ ⊂
Wδ ⊂ W ∗ to prove a priori energy estimates. By definition, it consists of allthe points in Wδ with time coordinate t ≤ t∗, i.e.,
W ∗δ = Wδ
⋂⎛
⎝⋃
−2≤t≤t∗�t
⎞
⎠ .
In the previous picture, the region W ∗δ is the part of the grey region below the
horizontal dash-dot line.For the purpose of future use, we introduce more notations to describe
various geometric objects.For each (t, u) ∈ [−2, t∗) × [0, δ], we use St,u to denote the closed two
dimensional surface
St,u := �t
⋂Cu . (2.13)
In particular, we have
W ∗δ =
⋃
(t,u)∈[−2,t∗)×[0,δ]St,u . (2.14)
123
S. Miao, P. Yu
For each (t, u) ∈ [−2, t∗) × [0, δ], we define
�ut :={(t, u′, θ) ∈ �t | 0 ≤ u′ ≤ u
},
Ctu :={(τ, u, θ) ∈ Cu | −2 ≤ τ ≤ t
},
Wtu :=
⋃
(t ′,u′)∈[−2,t)×[0,u]St ′,u′ .
(2.15)
One can consult the previous picture to visualize those objects. In particular,Wt
u is the grey region bounded by �ut and Ct
u .In what follows when working in Wt
u , we usually omit the superscript u to
write μum(t) as μm(t), whenever there is no confusion.
We define the vectorfield T in W ∗δ by the following three conditions:
1. T is tangential to �t ;2. T is orthogonal (with respect to g) to St,u for each u ∈ [0, δ];3. Tu = 1.
The letter T stands for “transversal” since the vectorfield is transversal to thefoliation of null hypersurfaces Cu .
In particular, the point (1) implies
T t = 0. (2.16)
According to (2.8)–(2.10), we have
Lu = 0, Lt = 1. (2.17)
In view of (2.10), (2.17), (2.16) and the fact Tu = 1, we see that the commu-tator
� := [L, T ] (2.18)
is tangential to St,u .In view of (2.8)–(2.10) and the fact Tu = 1 we have
g(L, T ) = −μ, g(L, L) = 0. (2.19)
Since T is spacelike with respect to g (indeed, �t is spacelike and T istangential to �t ), we denote
g(T, T ) = κ2, κ > 0. (2.20)
123
On the formation of shocks for quasilinear wave equations
Lemma 2.2 We have the following relations for L, T, μ and κ:μ = cκ, L = ∂0 − cκ−1T, (2.21)
where ∂0 is the standard time vectorfield in Minkowski spacetime.
Proof The vectorfield ∂0 is perpendicular to�t and therefore is perpendicularto St,u . Since L and T are two linearly independent vectorfields perpendicularto St,u and Lt = ∂0t = 1, we have
∂0 = L + f T
for some scalar function f . On the other hand, ∂0 is perpendicular to�t henceto T , we have
0 = g(∂0, T ) = g(L, T ) + f g(T, T ) = −μ + f κ2.
Therefore, f = μ
κ2= c
κand the second formula in (2.21) follows.
For the first formula, in view of the defining equation of the optical metricg, we have
−c2 = g(∂0, ∂0) = g(L + f T, L + f T ) = −2 f μ + f 2κ2.
Since f = cκ, we can solve for μ to complete the proof. �
Remark 2.3 On the initial Cauchy surface �−2, since u = r − 2, we haveT = ∂r and κ = 1. Therefore, by using the standard rectangular coordinates,we obtain that
L = ∂t − c∂r .
This is coherent with the notations and computations in Lemma 1.1.
2.2.3 The optical coordinates
We construct a new coordinate system onW ∗δ . If shocks form, the new coordi-
nate system is completely different from the rectangular coordinates. Indeed,we will show that they define two differentiable structures onW ∗
δ when shocksform.
Given u ∈ [0, δ], the generators of Cu define a diffeomorphism betweenS−2,u and St,u for each t ∈ [−2, t∗). Since S−2,u is diffeomorphic to thestandard sphere S
2 ⊂ R3 in a natural way. We obtain a natural diffeo-
morphism between St,u and S2. If local coordinates (θ1, θ2) are chosen on
S2, the diffeomorphism then induces local coordinates on St,u for every
123
S. Miao, P. Yu
(t, u) ∈ [−2, t∗) × [0, δ]. The local coordinates (θ1, θ2), together with thefunctions (t, u) define a complete system of local coordinates (t, u, θ1, θ2)for W ∗
δ . This new coordinates are defined as the optical coordinates.We now express for L , T and the optical metric g in the optical coordinates.First of all, the integral curves of L are the lines with constant u and θ . Since
Lt = 1, therefore in optical coordinates we have
L = ∂
∂t. (2.22)
Similarly, since Tu = 1 and T is tangential to �t , we have
T = ∂
∂u− � (2.23)
with � a vectorfield tangential to St,u . Locally, we can express � as
� =∑
A=1,2
�A ∂
∂θ A, (2.24)
The metric g then can be written in the optical coordinates (t, u, θ1, θ2) as
g = −2μdtdu + κ2du2 + /gAB(dθA + �Adu)(dθ B + �Bdu) (2.25)
with
/gAB = g
(∂
∂θ A,
∂
∂θ B
), 1 ≤ A, B ≤ 2. (2.26)
To study the differentiable structure defined by the optical coordinates,we study the Jacobian � of the transformation from the optical coordinates(t, u, θ1, θ2) to the rectangular coordinates (x0, x1, x2, x3).
First of all, since x0 = t , we have
∂x0
∂t= 1,
∂x0
∂u= ∂x0
∂θ A= 0.
Secondly, by (2.23), we can express T = T i∂i in the rectangular coordinates(x1, x2, x3) as
T i = ∂xi
∂u−∑
A=1,2
�A ∂xi
∂θ A
123
On the formation of shocks for quasilinear wave equations
In view of the fact that T is orthogonal to ∂∂θ A
with respect to the Euclideanmetric (which is the induced metric of g on �t !), we have
� = det
⎛
⎜⎝
T 1 T 2 T 3
∂x1
∂θ1∂x2
∂θ1∂x3
∂θ1
∂x1
∂θ2∂x2
∂θ2∂x3
∂θ2
⎞
⎟⎠ = ‖T ‖
∥∥∥∥
∂
∂θ1∧ ∂
∂θ2
∥∥∥∥ = c−1μ
√det /g,
where‖·‖measures themagnitude of a vectorfieldwith respect to theEuclideanmetric in R
3 (defined by the rectangular coordinates (x1, x2, x3)).We end the discussion by an important remark. We can also read the con-
clusions from the following picture:
Remark 2.4 (Geometric meaning of μ) In the sequel, we will show that thewave speed function c will be always approximately equal to 1 in W ∗
δ . Sinceμ = cκ , we may think of μ being κ in a efficient way.
On the other hand, by the definition of T , in particular Tu = 1, we knowthat κ−1 is indeed the density of the foliation by the Cu’s. This is because
g(T, T ) = κ2. Since the optical metric coincides with the Euclidean metricon each constant time slice�t , byμ ∼ κ , we arrive at the following conclusion:
• μ−1 measures the foliation of the incoming null hypersurfaces Cu’s.
Therefore, by regarding shock formation as the collapsing (i.e. the densityblows up) of the characteristics (� the incoming null hypersurfaces), we maysay that
• Shock formation is equivalent to μ → 0.
By virtue of the formula � = c−1μ√det /g, it is clear (the volume ele-
ment√det /g will be controlled in the sequel) that if shock forms then the
coordinate transformation between the optical coordinates and the rectangu-
123
S. Miao, P. Yu
lar coordinates will fail to be a diffeomorphism. Therefore, we can also saythat
• Shock formation is equivalent to the fact that the optical coordinates onthe maximal development defines a different differentiable structure (com-pared to the usual differentiable structure induced from the Minkowskispacetime).
2.3 Connection, curvature and structure equations
We use∇ to denote the Levi-Civita connection of g and use XA to denote ∂∂θ A
.
The 2nd fundamental form of the embedding St,u ↪→ Cu is
χAB
= g(∇XA L, XB). (2.27)
The trace/traceless part is defined by trχ = trg/ χ = g/ ABχAB
and χAB
=χ
AB− 1
2 trχ g/ AB . Let T = cμ−1T . Then, g(T , T ) = 1. The 2nd fundamentalform of St,u ↪→ �t is
θAB = g(∇XA T , XB) (2.28)
By virtue of (2.2), we have χAB
= −c θAB . Thanks to Gauss’ TheoremaEgregium, the Gauss curvature K of St,u is
K = 1
2(tr/gθ)
2 − 1
2|θ |2
/g = 1
2c−2((tr/gχ)
2 − |χ |2/g
). (2.29)
We introduce an outgoing null vectorfield
L = c−2μL + 2T (2.30)
so that g(L , L) = −2μ. The corresponding 2nd fundamental form is χAB =g(∇XA L , XB). Similarly, we define trχ = trg/ χ = g/ ABχAB and χAB =χAB − 1
2 trχ g/ AB .The torsion one forms η
Aand ζ
Aare defined by ζ
A= g(∇XA L, T ) and
ηA
= −g(∇XAT, L). They are related to the inverse density μ by ηA
=ζA
+ XA(μ) and ζ A= −c−1μ XA(c).
123
On the formation of shocks for quasilinear wave equations
The covariant derivative∇ is now expressed in the frame (T, L, X1, X2) by(∇/ is the restriction of ∇ on St,u)
∇L L = μ−1(Lμ)L, ∇T L = ηAXA − c−1L(c−1μ)L,
∇XA L = −μ−1ζAL + χ
AB XB, ∇LT = −ζ AXA − c−1L(c−1μ)L,
∇T T = c−3μ(T c + L(c−1μ))L + (c−1(T c + L(c−1μ))
+ T (log(c−1μ)))T − c−1μ/gAB XB(c−1μ)XA,
∇XAT = μ−1ηAT + c−1μθAB/g
BC XC , ∇L XA = ∇XA L, ∇XA XB
= ∇/ XAXB + μ−1χ
ABT .
In terms of null frames (L , L, X1, X2), we have
∇L L = −L(c−2μ)L + 2ηAXA, ∇L L = −2ζ AXA,
∇L L = (μ−1Lμ + L(c−2μ))L − 2μX A(c−2μ)XA,
∇XA L = μ−1ηAL + χA
B XB,
∇XA XB = ∇/ XAXB + 1
2μ−1χ
ABL + 1
2μ−1χABL.
In the Cartesian coordinates, the only non-vanishing curvature componentsare R0i0 j ’s:
R0i0 j = 1
2
d(c2)
dρ∇i∇ jρ + 1
2
d2(c2)
dρ2∇iρ∇ jρ − 1
4c−2
∣∣∣∣d(c2)
dρ
∣∣∣∣
2
∇iρ∇ jρ.
In the optical coordinates, the only nonzero curvature components are αAB =R(XA, L, XB, L):
αAB = 1
2
d(c2)
dρ/∇2XA,XB
ρ − 1
2μ−1 d(c
2)
dρT (ρ)χ
AB
+ 1
2
(d2(c2)
dρ2− 1
2c−2
∣∣∣∣d(c2)
dρ
∣∣∣∣
2)
XA(ρ)XB(ρ).
We define
α′AB = 1
2
d(c2)
dρ/∇2XA,XB
ρ + 1
2
[d2(c2)
dρ2− 1
2c−2
∣∣∣∣d(c2)
dρ
∣∣∣∣
2]
XA(ρ)XB(ρ).
123
S. Miao, P. Yu
Therefore, we write
αAB = −1
2μ−1 d(c
2)
dρT (ρ)χ
AB+ α′
AB . (2.31)
Remark 2.5 As a convention, we say that the first term on the right hand sideof (2.31) is singular in μ (since μ may go to zero). The second term α′
AB isregular in μ.
Indeed, in the course of the proof, wewill see thatα′AB are bounded andαAB
behaves exactly as μ−1 in amplitude. Therefore, in addition to two equivalentdescriptions of the shock formation in Remark 2.4, we have another geometricinterpretation:
• Shock formation is equivalent to the fact that curvature tensor of the opticalmetric g becomes unbounded.
Compared to the one dimensional picture of shock formation in conservationlaws, e.g., for inviscid Burgers equation, this new description of shock forma-tion is purely geometric in the following sense: it does not even depend on thechoice of characteristic foliation (because the curvature tensor is tensorial!).
In the frame (T, L, ∂∂θ A
), the connection coefficients and the curvature com-ponents satisfies the following structure equations:
L(χAB
) = μ−1(Lμ)χAB
+ χACχ
BC− αAB, (2.32)
/divχ − /dtrχ = −μ−1(ζ · χ − ζ trχ), (2.33)
/LTχ AB=(∇/ ⊗η)AB + μ−1(ζ ⊗η)AB−c−1L(c−1μ)χ
AB+ c−1μ(θ⊗χ)AB,
(2.34)
where (ζ · χ)B = /gACζAχBC
, (∇/⊗η)AB = 12 (∇/ AηB
+ ∇/ BηA), (ζ ⊗η)AB =
12 (ζ A
ηB
+ ζBηA) and (θ⊗χ)AB = 1
2 (θACχCB
+ θBCχCA). By taking the trace
of (2.32), we have
Ltrχ = μ−1(Lμ)trχ − |χ |2/g − trα. (2.35)
The inverse density function μ satisfies the following transport equation:
Lμ = m + μe, (2.36)
123
On the formation of shocks for quasilinear wave equations
with m = −12d(c2)dρ Tρ and e = 1
2c2d(c2)dρ Lρ. With these notations, we have
αAB = μ−1mχAB
+ α′AB .
Regarding the regularity in μ, we use (2.36) to replace Lμ in (2.32). Thisyields
L(χAB
) = eχAB
+ χACχ
BC− α′
AB . (2.37)
Compared to the original (2.32), the new equation is regular μ in the sensethat it has no μ−1 terms.
2.4 Rotation vectorfields
Although g∣∣�t
is flat, the foliation St,u is different from the standard spherical
foliations. In the Cartesian coordinates on �t , let 1 = x2∂3 − x3∂2, 2 =x3∂1 − x1∂3 and 3 = x1∂2 − x2∂1 be the standard rotations. Let � bethe orthogonal projection to St,u (embedded in �t ). The rotation vectorfieldsRi ∈ �(T St,u) (i = 1, 2, 3) are defined by
Ri = � i . (2.38)
Let indices i, j, k ∈ {1, 2, 3}. We use the T k , Lk and XkA to denote the com-
ponents for T , L and XA in the Cartesian frame {∂i } on �t (notice that L hasalso a 0 component L0 = 1). We use T = cμ−1T is the outward unit normalof St,u in�t . We introduce some functions to measure the difference betweenthe foliations St,u and the standard spherical foliations.
The functions λi ’s measure the derivation from Ri to i :
λi T = i − Ri . (2.39)
The functions y′k’s measure the derivation from T to the standard radial vec-torfield xi
r ∂i :
y′k = T k − xk
r. (2.40)
We also define (we will show that |yk − y′k | is bounded by a negligible smallnumber)
yk = T k − xk
u − t. (2.41)
123
S. Miao, P. Yu
The functions zk’s measure the derivation of L from ∂t − ∂r in Minkowskispacetime:
zk = Lk + xk
u − t= −(c − 1)xk
u − t− cyk .
Finally, the rotation vectorfields can be expressed as
Ri = i − λi
3∑
j=1
T j∂ j , λi =3∑
j,k,l=1
εilk xl yk, (2.42)
where εi jk is the totally skew-symmetric symbol.
3 Initial data, bootstrap assumptions and the main estimates
3.1 Preliminary estimates on initial data
In the Main Theorem, we take the so called short pulse datum for (�) on�δ−r0 .Recall that φ(−r0, x) = δ3/2φ0(
r−r0δ
, θ) and ∂tφ(−r0, x) = δ1/2φ1(r−r0δ
, θ),where φ0, φ1 ∈ C∞
0
((0, 1] × S
2). The condition (3) in the statement of the
Main Theorem reads as
‖Lφ‖L∞(�δ−2
) � δ3/2, ‖L2φ‖L∞(�δ−2
) � δ3/2.
We now derive estimates for φ and its derivatives on �δ−2. These estimatesalso suggest the estimates, e.g. the bootstrap assumptions in next subsection,that one can expect later on.
For φ and ψ = Aφ where A ∈ {∂α}, by the form of the data, we clearlyhave
‖φ‖L∞(�δ−2)� δ3/2, ‖ψ‖L∞(�δ−2)
� δ1/2. (3.1)
We will use Z or Z j to denote any vector from {T, Ri , Q} where Q = t L .On �δ−2, Z is simply ∂r , i or −r0(∂t − ∂r ), therefore, we have ‖Z1 ◦ Z2 ◦· · · ◦ Zm(ψ)‖L∞(�δ−2)
� δ1/2−l where l is the number of T ’s appearing in{Z j }1≤ j≤m . We shall use the following schematic expression
‖Zmψ‖L∞(�δ−2)� δ1/2−l, (3.2)
with l is the number of T ’s and Z ∈ {T, i , Q}. We remark that in this paperl ≤ 2 and Q appears at most twice in the string of Z ’s.
123
On the formation of shocks for quasilinear wave equations
We also consider the incoming energy for Zmψ on�δ−2. According to (3.2),we have
‖L(Zmψ)‖L2(�δ−2)+ ‖/d(Zmψ)‖L2(�δ−2)
� δ1−l, ‖T (Zmψ)‖L2(�δ−2)� δ−l
where /d denotes for the exterior differential on St,u . In terms of L , form ∈ Z≥0,we obtain
‖L(Zmψ)‖L2(�δ−2)+ ‖/d(Zmψ)‖L2(�δ−2)
� δ1−l, ‖L(Zmψ)‖L2(�δ−2)� δ−l
(3.3)
where l is the number of T ’s in Z ’s.We also consider the estimates on some connection coefficients on �−r0 .
For μ, since we have g(T, T ) = c−2μ2 and T = ∂r on �−r0 , we then have
μ = c on �r0 . Since c = (1 + 2G ′′(0)(∂tφ)2)−12 , according to (3.1), for
sufficiently small δ, we obtain
‖μ − 1‖L∞(�δ−2)� δ. (3.4)
For χAB
, since χAB
= −cθAB = − cr0 /gAB , we have χ AB
+ 1r0 /gAB = (1 −
c) 1r0 /gAB . Hence,
∥∥∥∥χ AB
+ 1
r0/gAB
∥∥∥∥L∞(�δ−2)
� δ. (3.5)
It measures the difference between the 2nd fundamental form with respect togαβ and mαβ .
3.2 Bootstrap assumptions and the main estimates
We expect the estimates (3.1)–(3.3) hold not only for t = −2 but also for latertime slice in W ∗
δ . For this purpose, we will run a bootstrap argument to derivethe a priori estimates for the Zmψ’s.
3.2.1 Conventions
We first introduce three large positive integers Ntop, Nμ and N∞. They will bedetermined later on. We require that Nμ = �34Ntop� and N∞ = �12Ntop� + 1.Ntop will eventually be the total number of derivatives applied to the linearizedequation �gψ = 0.
123
S. Miao, P. Yu
To count the number of derivatives, we define the order of an object. Thesolution φ is considered as an order −1 object. The variations ψ = Aφ are oforder 0. Themetric g depends only onψ , so it is of order 0. The inverse densityfunctionμ is of order 0. The connection coefficients are 1st order derivatives ong, hence, of order 1. In particular, χ
ABis of order 1. Let α = (i1, . . . , ik−1) be
a multi-index with i j ’s from {1, 2, 3}. We use Zαψ as a schematic expressionof Zi1Zi2 · · · Zik−1ψ . The order of Zαψ is |α|, where |α| = k−1. Similarly, forany tensor of order |α|, after taking m derivatives, its order becomes |α| +m.The highest order objects in this paper will be of order Ntop + 1.
Let l ∈ Z≥0 and k ∈ Z. We use Olk or O≤l
k to denote any term of order l orat most l with estimates
‖Olk‖L∞(�δ
t )� δ
12 k, ‖O≤l
k ‖L∞(�δt )
� δ12 k .
Similarly, we use �lk or �≤l
k to denote any term of order l or at most l withestimates
‖�lk‖L∞(�δ
t )� δ
12 k, ‖�≤l
k ‖L∞(�δt )
� δ12 k,
and moreover, it can be explicitly expressed a function of the variationsψ . Forexample, ∂tφ ·∂iφ ∈ �0
2 ; A term of the form∏n
i=1 Zαiψ so that max |αi | ≤ m
is �≤mn−2l , where l is the number of T appearing in the derivatives. Note that χ
and μ can not be expressed explicitly in terms of ψ .TheOl
k terms (or similarly the�lk terms) obey the following algebraic rules:
O≤lk + O≤l ′
k′ = O≤max(l,l ′)min(k,k′) , O≤l
k O≤l ′k′ = O≤max(l,l ′)
k+k′ .
3.2.2 Bootstrap assumptions on L∞ norms
Motivated by (3.2), we make the following bootstrap assumptions (B.1) onW ∗
δ : For all t and 2 ≤ |α| ≤ N∞,2
‖ψ‖L∞(�δt )
+ ‖Lψ‖L∞(�δt )
+ ‖/dψ‖L∞(�δt )
+ δ‖Tψ‖L∞(�δt )
+δl‖Zαψ‖L∞(�δt )
� δ12 M. (B.1)
where l is the number of T ’s appearing in Zα andM is a large positive constantdepending on φ. We will show that if δ is sufficiently small which may dependon M , then we can choose M in such a way that it depends only on the initialdatum.
2 For a multi-index α, the symbol α−1means another multi-index β with degree |β| = |α|−1.
123
On the formation of shocks for quasilinear wave equations
3.2.3 Energy norms
Let dμ/g be the volume form of /g, for a function f (t, u, θ), we define
∫
�ut
f =∫ u
0
(∫
St,u′f (t, u′, θ)dμ/g
)
du′,
∫
Ctu
f =∫ t
−r0
(∫
Sτ,uf (τ, u, θ)dμ/g
)
dτ.
For a function �(t, u, θ), we define the energy flux through the hypersur-faces �u
t and Ctu as
E(�)(t, u) =∫
�ut
(L�)2 + μ2|/d�|2, F(�)(t, u) =∫
Ctu
μ|/d�|2,
E(�)(t, u) =∫
�ut
μ(L�)2 + μ|/d�|2, F(�)(t, u) =∫
Ctu
(L�)2.
(3.6)
For each integer 0 ≤ k ≤ Ntop, we define
Ek+1(t, u) =∑
ψ
∑
|α|=k−1
δ2l E(Zα+1ψ)(t, u),
Fk+1(t, u) =∑
ψ
∑
|α|=k−1
δ2l F(Zα+1ψ)(t, u),
Ek+1(t, u) =∑
ψ
∑
|α|=k−1
δ2l E(Zα+1ψ)(t, u),
Fk+1(t, u) =∑
ψ
∑
|α|=k−1
δ2l F(Zα+1ψ)(t, u),
(3.7)
where l is the number of T ’s appearing in Zα . The symbol∑
ψ means to sumover all the first order variations Aφ of ψ . For the sake of simplicity, we shallomit this sum symbol in the sequel.
For each integer 0 ≤ k ≤ Ntop, we assign a nonnegative integer bk to k insuch a way that
Wecallbk’s the blow-up indices. The sequence (bk)0≤k≤Ntop will be determinedlater on.
123
S. Miao, P. Yu
For each integer 0 ≤ k ≤ Ntop, we also define the modified energy Ek(t, u)and Ek(t, u) as
Ek+1(t, u) = supτ∈[−r0,t]
{μum(τ )
2bk+1Ek+1(τ, u)},
Ek+1(t, u) = supτ∈[−r0,t]
{μum(τ )
2bk+1Ek+1(τ, u)}
Fk+1(t, u) = supτ∈[−r0,t]
{μum(τ )
2bk+1Fk+1(τ, u)},
Fk+1(t, u) = supτ∈[−r0,t]
{μum(τ )
2bk+1Fk+1(τ, u)}
(3.9)
We now state the main estimates of the paper.
Theorem 3.1 There exists a constant δ0 depending only on the seed data φ0and φ1, so that for all δ < δ0, there exist constants M0, Ntop and (bk)0≤k≤Ntop
with the following properties
• M0, Ntop and (bk)0≤k≤Ntop depend only on the initial datum.• The inequalities (B.1) holds for all t < t∗ with M = M0.• Either t∗ = −1 and we have a smooth solution in the time slab [−2,−1];or t∗ < −1 and then ψα’s as well as the rectangular coordinates xi ’sextend smoothly as functions of the coordinates (t, u, θ) to t = t∗ andthere is at least one point on �δ
t∗ where μ vanishes, thus we have shockformation.
• If, moreover, the initial data satisfies the largeness condition (1.6), thenin fact t∗ < −1.
Before we start the detailed analysis, it is instructional to provide a 3-stepscheme to illustrate the structure of the proof (of Theorem 3.1):
123
On the formation of shocks for quasilinear wave equations
3.3 Preliminary results based on (B.1)
3.3.1 Estimates on metric and connection
We start with bounds on c.
Lemma 3.2 For sufficiently small3 δ, we have
‖c − 1‖L∞(�δt )
� δM2,1
2≤ c ≤ 2,
‖Lc‖L∞(�δt )
+ ‖/dc‖L∞(�δt )
� δM2, ‖T c‖L∞(�δt )
� M2.
(3.10)
Proof Let ψ0 = ∂tφ. Since c = (1 + 3G ′′(0)(∂tφ)2
)− 12 , in view of (B.1),
we can take ε = 14G ′′(0)M2 . Therefore, the quantity in the parenthesis falls in
[14 , 74 ], this implies the bound on c.
On the other hand, Lc = −(3/2)G ′′(0)(1 + 3G ′′(0)ψ2
0
)− 32ψ0 · Lψ0. We
then use (B.1) exactly in the same way as above, this gives the bound on Lc.Similarly, we can obtain other bounds in (3.10). �
We now derive estimates on m, e and μ:
Lemma 3.3 For sufficiently small δ, we have
‖m‖L∞(�δt )
+ ‖/dm‖L∞(�δt )
� M2, ‖Tm‖L∞(�δt )
� δ−1M2, (3.11)
‖e‖L∞(�δt )
+ ‖/de‖L∞(�δt )
� δM2, ‖T e‖L∞(�δt )
� M2, (3.12)
‖μ‖L∞(�δt )
+ ‖Lμ‖L∞(�δt )
� M2. (3.13)
Proof Let ψ0 = ∂tφ. We first bound m. Since m ∼ (1 ± ψ20 )
−2ψ0 · Tψ0,according to (B.1), we have
‖m‖L∞(�δt )
� 1
(1 − 3G ′′(0)‖ψ0‖2L∞(�δt ))2
‖ψ0‖L∞(�δt )
· ‖Tψ0‖L∞(�δt )
� 1
(1 − 3G ′′(0)δM2)2· δ 1
2 M · δ− 12 M.
3 This sentence always means that, there exists ε = ε(M) so that for all δ ≤ ε, we have …
123
S. Miao, P. Yu
This clearly implies the bound on m for sufficiently small δ. For the deriva-tives of m, say Tm, it is based on (B.1) and the explicit computation Tm =
3G ′′(0)(1+3G ′′(0)ψ2
0 )2ψ0 · T 2ψ0 + 3G ′′(0)
(1+3G ′′(0)ψ20 )
2 (Tψ0)2 − 36G ′′(0)2
(1+3G ′′(0)ψ20 )
3ψ20 (Tψ0)
2.
Similarly, we can bound /dm.The estimates on e can be derived exactly in the same way, so we omit the
proof.To bound μ, we integrate the equation Lμ = m + μe to derive
μ(t, u, θ) = exp
(∫ t
−r0e(τ )dτ
)μ(−r0, u, θ)
+∫ t
−r0exp
(∫ t
τ
e(τ ′)dτ ′)m(τ, u, θ)dτ. (3.14)
Then (3.4), (3.11) and (3.12) imply the estimate on μ immediately. For Lμ,we simply combine Lμ = m + μe, (3.11), (3.12) and the bound on μ. Thiscompletes the proof. �
We move to the bounds on Tμ and /dμ:
Lemma 3.4 For sufficiently small δ, we have
‖Tμ‖L∞(�δt )
� δ−1M2, ‖/dμ‖L∞(�δt )
� M2. (3.15)
Proof The idea is to commute T and /d with Lμ = m + eμ. For /dμ, we have/LL(/dμ) = e/dμ + (/dm + μ/de), We have already seen that ‖/dm‖L∞(�δ
t )+
‖μ/de‖L∞(�δt )
� M2. Therefore, we integrate along L to derive the desiredbound on /dμ.
For Tμ, we have L(Tμ) = eTμ + (Tm + μT e − (ζ A + ηA)/d Aμ). We
first show that
‖ζ‖L∞(�δt )
� δM2, ‖η‖L∞(�δt )
� M2. (3.16)
In fact, ζA
= −c−1μ/d A(c). The bound on ζ the follows immediately from(3.10) and (3.13). Since η = ζ + /dμ, the bound on η is also clear because wehave just obtained estimates on /dμ.
Back to the formula for L(Tμ), for small δ, we bound the terms in theparenthesis by
‖(Tm + μT e − (ζ A + ηA)/d Aμ)‖L∞(�δ
t )� δ−1M2.
We then integrate to derive the bound for Tμ. This completes the proof. �
123
On the formation of shocks for quasilinear wave equations
We now estimate χAB
. For this purpose, we introduce
χ ′AB
= χAB
+ /gAB
u − t. (3.17)
which measures the deviation of χAB
from the null 2nd fundamental form inMinkowski space. We have
where all norms are defined with respect to /g.Let P(t) be the property that ‖χ ′‖L∞(�δ
t )≤ C0δM2 for all t ′ ∈ [−r0, t].
By choosing C0 suitably large, according to the assumptions on initial data,we have ‖χ ′‖L∞(�δ−2)
< C0δ. It follows by continuity that P(t) is true fort sufficiently close to −r0. Let t0 be the upper bound of t ∈ [−r0, t0] forwhich P(t) holds. By continuity, P(t0) is true. Therefore, for t ≤ t0, we have|χ ′|+|e| ≤ (C0+C1)δM2 for a universal constantC1.According to the explicitformula of α′ and (B.1), for sufficiently small δ, there is a universal constantC3so that ‖α′‖L∞(�δ
t )≤ C3δM2. In view of (6.20), there is a universal constant
C4 so that
L((t − u)2|χ ′|) ≤ C4(t − u)2((C0 + C1)δM
2|χ ′| + (C1 + C3)δM2).
(3.21)
If we define x(t) = (t − u)2|χ ′| along the integral curve of L , then we can
rewrite (3.21) as dxdt ≤ f x + g, where f (t) = C4(C0 + C1)δM2 and g(t) =
C4(C1 + C3)δM2. By integrating from −r0 to t , we obtain
x(t) ≤ e∫ t−r0
f (t ′)dt ′(x(−r0) +
∫ t
−r0e− ∫ t ′−r0
f (t ′′)dt ′′g(t ′)dt ′
).
123
S. Miao, P. Yu
Taking into account the facts that∫ t−r0
f (t ′)dt ′ ≤ C4(C0 + C1)δM2 and∫ t−r0
g(t ′)dt ′ ≤ C4(C1 +C3)δM2, since (t − u) ∼ 1 on the support of χ ′, forsome universal constant C5, we have
‖χ ′‖L∞(�δt )
≤ C5eC4(C0+C1)δM2(‖χ ′‖L∞(�δ−2)
+ C4(C1 + C3)δM2).
(3.22)
We then fix C0 in such a way that C0 > 2C5C4(C1 + C3) and C0 >‖χ ′‖
L∞(�δt )
4C5. Provided δ satisfying δ < log 2
C4(C0+C1)M2 , the estimate (3.22) implies
‖χ ′‖L∞(�δt )< C0δM2 for all t ∈ [−r0, t0]. By continuity,P(t) holds for some
t > t0. Hence the lemma follows. �
Remark 3.6 (Estimates related to the conformal optical metric g) We shall use˜ to indicate the quantities defined with respect to g, e.g., ∇ is the Levi-Civitaconnection of g and χ is the 2nd fundamental form of Su,t ↪→ Cu with respect
to L and g: χAB
= g(∇XA L, XB).We expect the quantities (with ) defined with respect to g have the similar
estimates as the counterparts (without ) definedwith respect to g. This is clear:the difference can be explicitly computed in terms of c and hence controlledby the estimates on c. For example, the difference between χ ′
ABand χ ′
ABis
χAB
− χAB
= − 12c2
L(c)/gAB , based on (3.10) and (3.18), we have
∥∥∥∥χ AB
+ /gAB
u − t
∥∥∥∥L∞(�δ
t )
� δM2. (3.23)
3.3.2 Estimates on deformation tensors
We use five vectorfields Z1 = T, Z2 = R1, Z3 = R2, Z4 = R3, Z5 = Q ascommutation vectorfields in the paper. We use Z to denote any possible choiceof the above Zi ’s. The notation Zα for a multi-index α = (i1, . . . , im) meansZi1Zi2 · · · Zim with i j ∈ {1, 2, 3, 4, 5}.We recall that, for Z , the deformation tensor (Z)π or (Z)π with respect to g
and g is defined by (Z)παβ = ∇αZβ +∇β Zα or (Z)παβ = 1c(Z)παβ + Z(1c )gαβ .
We now compute the deformation tensors with respect to g. The deformationtensor of Z1 = T is given by4
4 We emphasize that the trace tr is defined with respect to g.
123
On the formation of shocks for quasilinear wave equations
By virtue of the estimates already derived, for sufficiently small δ, we have
‖μ−1(T )πLL‖L∞(�δt )
� δ−1M2, ‖(T )πLL‖L∞(�δt )
� δ−1M2,
‖μ−1(T )πL A‖L∞(�δt )
� M2,
‖(T )πL A‖L∞(�δt )
� M2, ‖μ−1(T )/π AB‖L∞(�δt )
� δM2,
‖μ−1tr(T ) /π‖L∞(�δt )
� 1.
(3.25)
The deformation tensor of Z5 = Q is given by
(Q)πLL = 0, (Q)πLL = 4t L(c−2μ)(c−1μ) − 4c−3μ2,
(Q)πLL = −2t L(c−1μ) − 2c−1μ
(Q)πL A = 0, (Q)πL A = 2tc−1(ζA
+ ηA),
(Q)/π AB = 2tc−1χ
AB, tr
(Q)/π = 2c−1t tr/gχ .
(3.26)
By virtue of the estimates already derived, for sufficiently small δ, we have
‖μ−1(Q)πLL‖L∞(�δt )
� M2, ‖(Q)πLL‖L∞(�δt )
� M2,
‖(Q)πL A‖L∞(�δt )
� M2,
‖(Q)/π AB‖L∞(�δt )
� δM2, ‖tr(Q) /π‖L∞(�δt )
� 1.
(3.27)
Actually the estimate for tr(Q) /π can be improved more precisely. By (3.27):
‖tr(Q) /π‖L∞(�ut )
� 1 and the Eq. (4.1) togetherwith the bootstrap assumptions,we see the order of the former is also 1. While the estimate for the latter is a bitmore delicate. Let us rewrite the following component of deformation tensorof Q:
tr(Q)
/π = 2c−1t tr/gχ = 2c−1t tr/gχ′ + 4
c−1u
u − t− 4(c−1 − 1) − 4
123
S. Miao, P. Yu
This tells us:
‖tr(Q) /π + 4‖L∞(�δt )
� δ. (3.28)
We study the deformation tensors of Ri ’s. Based on (2.42), we compute(Ri )παβ with respect to g:
(Ri )πLL = 0, (Ri )πT T = 2c−1μ · Ri (c−1μ),
(Ri )πLT = −Ri (μ),(Ri )πAB = −2λiθAB ,
(Ri )πT A = −c−1μ
(θAB − /gAB
u − t
)Ri
B + c−1μεik j yk X A
j + λi X A(c−1μ),
(Ri )πL A = −χAB
RiB + Lkεik j X A
j + cμ−1λiζ A
= −(χ
AB+ /gAB
u − t
)Ri
B + εik j zk X A
j + cμ−1λiζ A.
(3.29)
The Latin indices i, j, k are defined with respect to the Cartesian coordinateson�t . To bound deformation of Ri , it suffices to control the λi ’s, yi ’s and zi ’s.
First of all, we have
|T i | + |Li | � 1. (3.30)
The proof is straightforward: g∣∣�δtis flat and T is the unit normal of St,u in
�δt , so |T i | ≤ 1. In the Cartesian coordinates (t, x1, x2, x3), L = ∂t − cT i∂i ,
so |Li | � 1.
Let r = (∑3
i=1 xi )
12 . Since Tr = c−1μ
∑3i=1
xi T i
r , (3.30) implies that|Tr | � M2. We then integrate from 0 to u, since r = −t when u = 0 and|u| ≤ δ, we obtain |r + t | � δM2. In application, for sufficiently small δ, weoften use r ∼ |t |. The estimate can also be written as
∣∣∣∣1
r− 1
u + |t |∣∣∣∣ � δM2. (3.31)
To control λi , we consider its L derivative. By definition λi = g( i , T ),we can write its derivative along L as Lλi = ∑3
k=1( i )k LT k =∑3
k=1( i )k X A(c)XA
k . As |t | ∼ r , we have | i | � |t | � 1, this implies
‖Lλi‖L∞(�δt )
� δM2. (3.32)
Since λi = 0 on �δ−r0 , we have
‖λi‖L∞(�δt )
� δM2. (3.33)
123
On the formation of shocks for quasilinear wave equations
To control yi ’s and zi ’s, let y = (y1, y2, y3) and x = (x1, x2, x3), we thenhave
∣∣∣∣y −
(1
r− 1
u − t
)x
∣∣∣∣
2
= |(g(T , ∂r ) − 1)∂r |2 + 1
r2
3∑
i=1
λ2i (3.34)
On the other hand, we have 1 − |g(T , ∂r )|2 = 1r2∑3
i=1 λ2i � δM2. While on
St,0, Since g(∂r , T ) = 1 on St,0, for sufficiently small δ, the angle between ∂rand T is less than π
2 , which implies 1 + g(T , ∂r ) ≥ 1. Therefore,
|1 − g(T , ∂r )| � δM2. (3.35)
Together with (3.33) and (3.34), this implies
|yi | � δM2, |y′i | � δM2. (3.36)
We then control zi from its definition
|zi | � δM2. (3.37)
The derivatives of λi on�t are given by XA(λi ) = (θAB− /gABu−t )R
Bi −εik j yk X
jA
and T (λi ) = −Ri (c−1μ). Hence,
‖/dλi‖L∞(�δt )
� δM2, ‖Tλi‖L∞(�δt )
� M2
Finally, we obtain the following estimates for the deformation tensor of Ri :5
(Ri )πLL = 0, ‖(Ri )πLT ‖L∞(�δt )
� M2, ‖μ−1(Ri )πT T ‖L∞(�δt )
� M2,
‖(Ri )πL A‖L∞(�δt )
� δM2, ‖(Ri )πT A‖L∞(�δt )
� δM4,
‖(Ri ) /π AB‖L∞(�δt )
� δ2M4, ‖tr(Ri ) /π‖L∞(�δt )
� δM4.
We use the relation L = c−2μL + 2T to rewrite the above estimates in nullframe as follows:
‖(Ri )πLL‖L∞(�δt )
� M2, ‖μ−1(Ri )πLL‖L∞(�δt )
� M2,
‖(Ri )πL A‖L∞(�δt )
� δM2, ‖(Ri )πL A‖L∞(�δt )
� δM4,
‖(Ri ) /π AB‖L∞(�δt )
� δ2M4, ‖tr(Ri ) /π‖L∞(�δt )
� δM4.
(3.38)
5 We emphasize that the traceless part of /π AB is defined with respect to /g.
123
S. Miao, P. Yu
The deformation tensors of Ri ’s with respect to g are estimated by
(Ri )πLL = 0, ‖(Ri )πLL‖L∞(�δt )
� M2, ‖μ−1(Ri )πLL‖L∞(�δt )
� M2,
‖(Ri )πL A‖L∞(�δt )
� δM2, ‖(Ri )πL A‖L∞(�δt )
� δM4,
‖(Ri )/π AB‖L∞(�δt )
� δ2M4, ‖tr(Ri ) /π‖L∞(�δt )
� δM4.
(3.39)
3.3.3 Applications
As in [5,8], we are able to show that the Ri derivatives are equivalent to the /dand ∇/ derivative. For a 1-form ξ on St,u , we have
∑3i=1 ξ(Ri )
2 = r2(|ξ |2 −(ξ(y′))2). This is indeed can be derived from the formula
∑3i=1(Ri )
a(Ri )b =
r2(δcd − y′c y′d)�ac�
bd , where a, b, c, d ∈ {1, 2, 3}. In view of (3.31), (3.36)
and the definition of y′i , for sufficiently small δ, we have∑3
i=1 ξ(Ri )2 ∼
r2|ξ |2. Since r is bounded below and above by a constant, we obtain∑3i=1 ξ(Ri )
2 ∼ |ξ |2. Similarly, for a k-covariant tensor ξ on St,u , we have∑3
i1,i2,...,ik=1 ξ(Ri1, Ri2, . . . , Rik )2 ∼ |ξ |2. In particular, we can take ξ = /dψ ,
therefore,∑3
i=1(Riψ)2 ∼ |/dψ |2. Henceforth, we omit the summation and
write schematically as |Riψ | ∼ |/dψ |.We can also compare the Ri -derivatives with the ∇/ -derivatives for tensors.
For St,u-tangential 1-form ξ and vectorfield X , let /LRi ξ be the orthogonalprojection of the Lie derivativeLRi ξ onto the surface St,u . Since (/LRi ξ)(X) =(∇/ Ri ξ)(X) + ξ(∇/ X Ri ), we obtain
3∑
i=1
|/LRi ξ |2 =3∑
i=1
|ξ(Ri )|2 + 23∑
i=1
ξ k(∇/ Ri ξ)a(∇/ Ri )ak
+3∑
i=1
ξ kξ l(∇/ Ri )ak(∇/ Ri )al .
We also have∑3
i=1 |∇/ Ri ξ |2 = r2(δcd − y′c y′d)(∇/ξ)ca(∇/ξ)da . In view of theestimates on y′i , for sufficiently small δ, we obtain
3∑
i=1
|∇/ Ri ξ |2 � |∇/ξ |2.
Let εi jk be the volume form on �t and vi be a St,u 1-form with rectan-gular components (vi )a = �b
aεibkξk . By virtue of the formula ( /∇Ri )kl =
�mk �
nl εinm − λiθkl , we have
123
On the formation of shocks for quasilinear wave equations
3∑
i=1
ξ kξ l(∇/ Ri )ak(∇/ Ri )al = |ξ |2 + 2c
3∑
i=1
λiξ · χ · vi + c2|χ · ξ |23∑
i=1
λ2i .
In view of the estimates on λi , for sufficiently small δ, we have∑3
i=1 ξkξ l
(∇/ Ri )ak(∇/ Ri )al ∼ |ξ |2. Similarly, we have
∣∣∑3i=1 ξ
k(∇/ Ri ξ)a(∇/ Ri )bk∣∣ �
|ξ ||∇/ξ |. Finally, we conclude that
|ξ |2 + |∇/ξ |2 �3∑
i=1
|/LRi ξ |2 � |ξ |2 + |∇/ξ |2.
Henceforth, we omit the summation and write schematically as |/LRi ξ | ∼|ξ | + |∇/ξ |. Similarly, for a tracefree symmetric 2-tensors θAB tangential toSt,u , we have |θ | + |∇/θ | � |/LRi θ | � |θ | + |∇/θ |. This will be applied toθ = χ
ABlater on.
Another application of the pointwise estimates based on the bootstrapassumption is to give an estimate on
√det /g. On �δ−2, S−2,u is around sphere
and√det /g is bounded above and below by positive absolute constants. On the
other hand, (3.17) and (3.18) implies that trχ is bounded. This together withthe formula
L(log√det /g
) = trχ (3.40)
gives the fact that√det /g is bounded and never vanishes along each null
generator.
3.3.4 Sobolev inequalities and elliptic estimates
To obtain the Sobolev inequalities on St,u , we introduce
I (t, u) = supU∈St,u ,∂U is C1
min(|U |, |St,u −U |)
|∂U |2
the isoperimetric constant on St,u , where |U |, |St,u − U | and |∂U | are themeasures of the corresponding sets with respect to /g on St,u . Therefore, inview of the fact that Ri ∼ ∇/ , for sufficiently small δ, we have the followingSobolev inequalities:
123
S. Miao, P. Yu
‖ f ‖W 1,4(St,u) � |I (t, u)| 14 |St,u |− 34(‖ f ‖L2(St,u) + ‖Ri f ‖L2(St,u)
+ ‖Ri R j f ‖L2(St,u)
),
‖ f ‖L∞(St,u) � |I (t, u)| 34 |St,u |− 12(‖ f ‖L2(St,u)
+ ‖Ri f ‖L2(St,u) + ‖Ri R j f ‖L2(St,u)
).
(3.41)
where ‖ f ‖W 1,4(St,u) is defined as ‖ f ‖W 1,4(St,u) = |St,u |−1/2‖ f ‖L4(St,u) +‖/d f ‖L4(St,u). It remains to control the isoperimetric constant I (t, u).
We use T to generate a diffeomorphism of St,u to St,0 which maps U ,St,u − U and ∂U to corresponding sets Uu , St,0 − Uu and ∂Uu on St,0. LetUu′, St,u′−Uu′, ∂Uu′ be the inverse images of these on each St,u′ foru′ ∈ [0, u].Since /LT g/ AB = 2c−1μθ = −2c−2μχ
AB, for u′ ∈ [0, u], we obtain
d
du′(|Uu′ |) = −
∫
Uu′c−2μtrχdμg/ ,
d
du′(|∂Uu′ |) = −
∫
Uu′c−2μχ(ν, ν)ds,
where ν is the unit normal of ∂Uu′ in St,0 and ds the element of arc lengthof ∂Uu′ . In view of the estimates on χ and μ derived before, for sufficientlysmall δ, we have
d
du′(|Uu′ |) � |Uu′ |, d
du′(|∂Uu′ |) � |∂Uu′ |.
Therefore, by integrating from 0 to u, we have
|U | ∼ |Uu|, |∂U | ∼ |∂Uu|.
Hence, I (t, u) ∼ I (t, 0) ∼ 1. Finally, since |St,u| ∼ 1 (This is seen by thefact dμ/g(t,u) ∼ dμ/g(−r0,0), which can be shown by calculations in [5].), weconclude that
‖ f ‖W 1,4(St,u) + ‖ f ‖L∞(St,u) � ‖ f ‖L2(St,u) + ‖Ri f ‖L2(St,u)
+ ‖Ri R j f ‖L2(St,u).(3.42)
We remark that, similarly, we have
‖ f ‖L4(St,u) � ‖ f ‖L2(St,u) + ‖Ri f ‖L2(St,u). (3.43)
We also have the following elliptic estimates for traceless two-tensors.
123
On the formation of shocks for quasilinear wave equations
Lemma 3.7 If δ is sufficiently small, for any traceless 2-covariant symmetrictensor θAB on St,u, we have
∫
St,uμ2(|∇/θ |2 + |θ |2)dμ/g �
∫
St,uμ2| /divθ |2 + |/dμ|2|θ |2/dμ/g (3.44)
Proof Let J A = θ BC∇/ BθAC − θ A
C( /divθ)C , then |J | � |θ |(| /∇θ | + | /divθ |).The Bochner formula says
|∇/θ |2 + 2K |θ |2 = 2| /divθ |2 + /divJ,
where K is the Gauss curvature. According to (2.29) and the estimates onχ
AB, for sufficiently small δ, we know that |K | ∼ 1. Therefore, we have
|∇/θ |2 + |θ |2 ∼ 2| /divθ |2 + /divJ,
We thenmultiply both sides byμ2 and integrate on St,u . The Cauchy–Schwarzinequality together with the above estimates on |J | as well as the divergencetheorem yields the inequality. �
4 The behavior of the inverse density function
The behavior of the inverse density function μ plays an dominant rôle in thispaper. The method of obtaining estimates on μ is to relate μ to its initial valueon�t=−r0 . Since the metric g depends only onψ0 = ∂tφ,μ is also determinedby ψ0. This leads naturally to the study of the wave equation �gψ0 = 0. Wecan rewrite it in the null frame as
L(Lψ0) + 1
2trχ · Lψ0
+(
−μ�/ψ0 + 1
2trχ · Lψ0 + 2ζ · /dψ0 + μ/d log(c) · /dψ0
)= 0. (4.1)
4.1 The asymptotic expansion for μ
We start with a lemma which relates Lψ0(t, u, θ) to its initial value on �−r0 .
Lemma 4.1 For sufficiently small δ, we have
∣∣|t |Lψ0(t, u, θ) − r0Lψ0(−r0, u, θ)
∣∣ � δ
12 M3. (4.2)
123
S. Miao, P. Yu
Proof We regard (4.1) as a transport equation for Lψ0. According to (B.1)and the estimates from previous sections, the L∞ norm of the terms in the big
parenthesis in (4.1) is bounded by δ12 M3. Hence,
∣∣∣∣L(Lψ0)(t, u, θ) + 1
2trχ(t, u, θ) · Lψ0(t, u, θ)
∣∣∣∣ � δ
12 M3.
Byvirtue of (3.23), this implies |L(Lψ0)(t, u, θ)− 1
u−t Lψ0(t, u, θ)| � δ12 M3.
Therefore, we obtain
|L((u − t)Lψ0)(t, u, θ)| � δ
12 M3.
Since |u| ≤ δ, we integrate from −r0 to t and this yields the desired estimates.�
Remark 4.2 The estimates 4.2 also hold for Ri Lψ0 or Ri R jψ0, e.g., see (4.12).To derive these estimates, we commute Ri ’s with (4.1) and follow the sameway as in the above proof.
Since L = c−2μL + 2T , as a corollary, we have
Corollary 4.3 For sufficiently small δ, we have
||t |Tψ0(t, u, θ) − r0Tψ0(−r0, u, θ)| � δ12 M3, (4.3)
||t |ψ0(t, u, θ) − r0ψ0(−r0, u, θ)| � δ32 M3. (4.4)
Proof For (4.4), we integrate (4.3) for u′ from 0 to u and use the fact thatψ0(t, 0, θ) = 0. �
We turn to the behavior of Lμ.
Lemma 4.4 For sufficiently small δ, we have
||t |2Lμ(t, u, θ) − r20 Lμ(−r0, u, θ)| � δM4. (4.5)
Proof According to (2.36), we write |t |2Lμ(t, u, θ) − r20 Lμ(−r0, u, θ) as
(|t |2m(t, u, θ) − r20m(−r0, u, θ))
+[|t |2 (μ · e)(t, u, θ) − r20 (μ · e)(−r0, u, θ)].
In view of (3.12), we bound the terms in the bracket by δM4 up to a universalconstant. Therefore,
|t |2Lμ(t, u, θ) − r20 Lμ(−r0, u, θ)
= (|t |2m(t, u, θ) − r20m(−r0, u, θ))+ O(δM4).
123
On the formation of shocks for quasilinear wave equations
Since
|t |2m(t, u, θ) − r20m(−r0, u, θ)
= τ 23G ′′(0)(ψ0(τ, u, θ) · Tψ0(τ, u, θ))
(1 + 3G ′′(0)ψ20 (τ, u, θ))
2
∣∣∣∣
τ=t
τ=−r0
.
It is clear that the estimates follow immediately after (4.3) and (4.4). � We are now able to prove an accurate estimate on μ.
Proposition 4.5 For sufficiently small δ, we have∣∣∣∣μ(t, u, θ) − 1 + r20
(1
t+ 1
r0
)Lμ(−r0, u, θ)
∣∣∣∣ � δM4. (4.6)
In particular, we have μ ≤ C0 where C0 is a universal constant dependingonly on the initial data.
Proof According to the previous lemma, we integrate Lμ:
μ(t, u, θ) − μ(−r0, u, θ) =∫ t
−r0Lμ(τ, u, θ)dτ =
∫ t
−r0
τ 2Lμ(τ, u, θ)
τ 2dτ
(4.5)=∫ t
−r0
r20 Lμ(−r0, u, θ)
τ 2+ O(δM4)
τ 2dτ
= −(1
t+ 1
r0
)r20 Lμ(−r0, u, θ) + O(δM4).
Therefore, we can use (3.4) to conclude. � We are ready to derive two key properties of the inverse density function
μ. The first asserts that the shock wave region is trapping for μ, namely, oncep ∈ Wshock , then all the points after p along the incoming null geodesic stayin Wshock .
Proposition 4.6 For sufficiently small δ and for all (t, u, θ) ∈ Wshock, wehave
Lμ(t, u, θ) ≤ − 1
4|t |2 � −1. (4.7)
Proof For (t, u, θ) ∈ Wshock , we have μ(t, u, θ) < 110 . In view of (4.6), we
claim that r20 Lμ(−r0, u, θ) < 0. Otherwise, since 1t + 1
r0< 0, we would
have μ(t, u, θ) ≥ 1 + O(δM2) > 110 , provided δ is sufficiently small. This
contradicts the fact that μ(t, u, θ) < 110 .
123
S. Miao, P. Yu
We can also use this argument to show that (1t + 1r0)r20 Lμ(−r0, u, θ) ≥ 1
2 .
Otherwise, for sufficiently small δ, we would have μ(t, u, θ) ≥ 12 + O(δM2)
> 110 .Therefore, we obtain r20 Lμ(−r0, u, θ) ≤ 1
2r0tr0+t . In view of (4.5), we have
t2Lμ(t, u, θ) ≤ 1
2
r0t
r0 + t+ O(δM2).
Here wewriteM2 instead ofM4 because we already know thatμ ≤ C0, whereC0 depends only on initial data. By taking a sufficiently small δ and noticingthat r0t
r0+t is bounded from above by a negative number, this yields the desiredestimates. � Remark 4.7 In the case when shock forms, i.e.μ → 0, by the previous propo-sition and (2.36), m = −1
2dc2dρ Tρ � −1. In other words,
Tρ ≥ c0 > 0
for some absolute constant c0.On the other hand, Tρ = cμ−1Tρ and ‖T ‖ = 1,therefore as μ → 0, ∇ρ blows up, so does ∇∂tφ.
Remark 4.8 We compare the estimates (3.13) and (4.5). (3.13) is rough:|Lμ| � M2; (4.5) is precise: |Lμ| ≤ C0 + δM2, where C0 depends onlyon the initial data. The improvement comes from integrating the wave equa-tion �gψ0 = 0 or equivalently (4.1).
4.2 The asymptotic expansion for derivatives of μ
We start with an estimate on derivatives of trχ .
Lemma 4.9 For sufficiently small δ ≤ ε, we have
‖Ltrgχ‖L∞ � M2, (4.8)
‖/dtrχ‖L∞ � δM2. (4.9)
Proof Wederive a transport equation for Riχ′AB
by commuting Ri with (3.19):
L(Riχ′AB
) = [L, Ri ]χ ′AB
+ eRiχ′AB
+ 2χ ′AC Riχ
′BC
+(Rie) · χ ′AB
− e
u − tRi /gAB − Ri
(e
u − t
)
/gAB − Riα′AB .
123
On the formation of shocks for quasilinear wave equations
Since [L, Ri ]A = (Ri )πLA, the commutator term [L, Ri ]χ ′
ABcan be bounded
by the estimates on the deformation tensors. We then multiply both sides byRiχ
′AB
and repeat the procedure that we used to derive (3.18). Since it isroutine, we omit the details and only give the final result
∥∥∥∥Ri
(χ
AB+ /gAB
u − t
)∥∥∥∥L∞
� δM2. (4.10)
In particular, this yields ‖Ri trχ‖L∞ � δM2 which is equivalent to (4.9).We derive a transport equation for Lχ ′
ABby commuting L with (3.19):
L(Lχ ′AB
) = eLχ ′AB
+ 2χ ′AC Lχ ′
BC+ Leχ ′
AB− e
u − tL/gAB
+ L
(e
u − t
)
/gAB − Lα′AB + L(c−2μ)Lχ ′
AB
+ /gCD(ζC
+ ηD)XC (χ
′AB
).
We then use Gronwall to derive∥∥∥∥L(χ
AB+ /gAB
u − t
)∥∥∥∥L∞
� M2. (4.11)
In particular, this yields ‖Ltrχ‖L∞ � M2. This is equivalent to (4.8). � We now derive estimates for Riψ0.
Lemma 4.10 For sufficiently small δ, we have
∣∣|t |LRiψ0(t, u, θ) − r0LRiψ0(−r0, u, θ)∣∣ � δ
12 M3. (4.12)
Proof We commute Ri with (4.1) and we obtain that L(Ri Lψ0) + 12 trχ ·
Ri Lψ0 = N with
N = Ri
(μ�/ψ0 − 1
2trχ · Lψ0 − 2ζ · /dψ0 − μ/d log(c) · /dψ0
)
−1
2Ri trχ · Lψ0 + [L, Ri ]Lψ0.
According to (B.1) and the previous lemma, N is bounded by δ12 M3. Hence,∣
∣L(Ri Lψ0) + 12 trχ · Ri Lψ0
∣∣ � δ
12 M3. We then integrate to derive
∣∣|t |Ri Lψ0(t, u, θ) − r0Ri Lψ0(−r0, u, θ)
∣∣ � δ
12 M3.
123
S. Miao, P. Yu
The commutator [Ri , L]ψ0 is bounded by δM3 thanks to the estimates ondeformation tensors. This completes the proof. �
We can obtain a better estimate for /dμ. The idea is to bound for Riμ anduse the fact that |/dμ| ∼ |Riμ|.Lemma 4.11 For sufficiently small δ, we have
‖/dμ‖L∞(�t ) � 1 + δM4. (4.13)
Proof We commute Ri with Lμ = m + eμ to derive
L(Riμ) = Rim + (eRiμ + μRie + [L, Ri ]μ).
According to (B.1) and the estimates on LRiψ0 (needed to bound Rim) fromthe previous lemma, it is straightforward to bound the terms in the parenthesisby δM2. Similar to (4.5), we obtain
||t |2L(Riμ)(t, u, θ) − |r0|2L(Riμ)(−r0, u, θ)| � δM4. (4.14)
Since ‖[L, Ri ]‖L∞ � δM2, we bound Riμ as
Riμ(t, u, θ) − Riμ(−r0, u, θ)(4.14)=
∫ t
−r0
r20 L(Riμ(r0, u, θ))
τ 2+ O(δM4)dτ
= −(1
t+ 1
r0
)r20 L(Riμ(−r0, u, θ)) + O(δM4).
This inequality yields (4.13) for sufficiently small δ. � We can also obtain a better estimate for Lμ.
Since the exact numeric constants and signs for the coefficients are irrelevantfor estimates, we replace all of them by 1 in the above expressions.
Since ζA
= −c−1μXA(c), by applying L and using (4.15), we obtain
|Lζ | � M2. Therefore, according (4.8), (B.1) and the estimates derived pre-viously in this section, we can bound all the terms on the right hand side andwe obtain
123
S. Miao, P. Yu
∣∣∣∣L(L
2ψ0) + 1
2trχ · L2ψ0
∣∣∣∣ � δ− 1
2 M3.
We then integrate from −r0 to t to obtain (4.17). � We can commute L twice with (3.19) or with (4.1) to obtain following
estimates.
Lemma 4.14 For sufficiently small δ, we have
‖L2trgχ‖ � M2δ−1, (4.19)
||t |L3ψ0(t, u, θ) − r0L3ψ0(−r0, u, θ)| � δ− 3
2 M3. (4.20)
We omit the proof since it is routine. Similarly, we commute L twice withLμ = m + eμ, we can use (4.19) and (4.20) to obtain
Lemma 4.15 There exists ε = ε(M) so that for all δ ≤ ε, we have
‖L2μ‖L∞(�t ) � δ−2 + δ−1M2, (4.21)
‖T 2μ‖L∞(�t ) � δ−2 + δ−1M2. (4.22)
We turn to another key property ofμwhich reflects the behavior ofμ−1Tμ.(4.5) suggests that if shock forms before t = −1, Lμ to behave as a constantnear s∗. Hence, μ is proportional to |t − s∗|. In view of (3.15), we expectμ−1Tμ behaves as |t − s∗|−1δ−1 near shocks (It is not integrable in t). Thenext proposition suggest a much better bound for μ−1(Tμ)+ by affording one
more derivative in T and we can improve |t − s∗|−1 to |t − s∗|− 12 .
Proposition 4.16 For p = (t, u, θ) ∈ Wδ, let (μ−1Tμ)+ be the nonnegativepart of μ−1Tμ. For sufficiently small δ and for all p ∈ Wshock, we have
(μ−1Tμ)+(t, u, θ) � 1
|t − s∗| 12δ−1. (4.23)
Proof We use a maximal principle type argument. Let γ : [0, δ] → W ∗δ be the
integral curve of T through the point p. Wemay choose the θ coordinates to beconstant along γ on the given�t and study the function f (u) = T (log(μ))(u).We assume f (u) attains its maximum at a point u∗ ∈ [0, δ]. We may alsoassume that this maximum is positive (Otherwise, (4.23) is automaticallytrue). Since u∗ is a maximum point, we have d
du ( f )(u∗) ≥ 0. In other
words, T (T (log(μ)))(u∗) ≥ 0. Therefore, at the point (t, u∗, θ), we have
123
On the formation of shocks for quasilinear wave equations
μ−1T 2μ − 1μ2 (Tμ)
2 ≥ 0. Hence,
‖μ−1(Tμ)+‖L∞([0,δ]) ≤√
‖T 2μ‖L∞([0,δ])infu∈[0,δ] μ(u)
. (4.24)
It suffices to bound the denominator and the numerator on the righthand side.For T 2μ, according to (4.22), we have
‖T 2μ‖L∞([0,δ]) � 1
|t |δ−2 � δ−2. (4.25)
For inf μ, we can assume (t, u, θ) ∈ Wshock . Otherwise we have the lowerbound μ ≥ 1
10 . According to (4.7), the condition (t, u, θ) ∈ Wshock implies|t |2(Lμ)(t, u, θ) ≤ −1
4 . Therefore, by (4.5), we have
r20 Lμ(−r0, u, θ) = t2Lμ(t, u, θ) + O(δM2) ≤ −1
4+ O(δM2) ≤ −1
6
provided that δ is sufficiently small. We now integrate from t to s∗ to derive
μ(t, u, θ) = μ(s∗, u, θ) −∫ s∗
tLμ(τ, u, θ) ≥ −
∫ s∗
tLμ(τ, u, θ)
= −∫ s∗
t
(r20 Lμ(−r0, u, θ)
|τ |2 + O(δM2)
)
≥ −r20 Lμ(−r0, u, θ)1
|s∗||t | |t − s∗| + O(δM2)|t − s∗|.
Therefore, for sufficiently small δ, we obtain |t − s∗| � μ. Together with(4.24) and (4.25), this completes the proof. �
5 The mechanism for shock formation
In this section, we assume Theorem 3.1 stated in Sect. 3.2 and we use knowl-edge on μ from last section to analyze a mechanism of shock formation. Inparticular, we have to use condition (1.6) in the Main Theorem. This is theonly place in the paper where we use (1.6).
The transport equation Lμ = m+μe is responsible for the shock formation.We first give precise bounds on each term on the righthand side. Since m =−1
2ddρ (c
2)Tρ, we havem = 3G ′′(0)(1+3G ′′(0)ψ2
0
)2ψ0 ·Tψ0. In view of (4.3), up to an
error of size δ12 , we can replace Tψ0(t, u, θ) by
r0|t |Tψ0(−r0, u, θ); in view of
123
S. Miao, P. Yu
(4.4), up to an error of size δ32 , we can replace ψ0(t, u, θ) by
r0|t |ψ0(−r0, u, θ).Therefore, we obtain
m = 3G ′′(0)r20|t |2ψ0(−r0, u, θ)Tψ0(−r0, u, θ) + O(δ)
= 3r20|t |2
(G ′′(0)φ1
(r − r0δ
, θ
)∂rφ1
(r − r0δ
, θ
))+ O(δ).
Since e is of size δ and |μ| � 1, we regardμe as an error term.We then obtain
Lμ(t, u, θ) = 3r20|t |2
(G ′′(0)φ1
(r − r0δ
, θ
)∂rφ1
(r − r0δ
, θ
))+ O(δ).
We then integrate this equation and we obtain
μ(t, u, θ) − μ(−r0, u, θ)
= 3∫ t
−r0
r20|τ |2 dτ ·
(G ′′(0)φ1
(r − r0δ
, θ
)∂rφ1
(r − r0δ
, θ
))+ O(δ)
= 3r20
(1
|t | − 1
r0
)(G ′′(0)φ1
(r − r0δ
, θ
)∂rφ1
(r − r0δ
, θ
))+ O(δ)
Since |μ(−r0, u, θ)−1| � δ, according to condition (1.6),we have (recall thatr0 = 2)
μ(t, u, θ) ≤ 1 − 3 · 22( 1|t | − 1
2
)16 + O(δ) = 1 − 2
(1|t | − 1
2
)+ O(δ)
Therefore, for sufficiently small δ, t can not be greater than −1, otherwise μwould be negative. In other words, shock forms before t = −1.
Corollary 5.1 If we introduce the vectorfield T := cμ−1T, then when shockforms, the second derivative of φ, T i∂i∂tφ, blows up.
Proof When shock forms, μ −→ 0, which means μ < 110 . Then by (4.7),
Lμ is negative and bounded from above. In other words, there is an absolutepositive constant C such that |Lμ| ≥ C . While from the propagation equationLμ = m + μe and the pointwise estimates |e| � δ, |μ| � 1, the estimate
|m| ≥ C follows if δ is sufficiently small. By the definition m = −12d(c2)dρ Tρ
and T = cμ−1T , the derivative of ρ, Tρ = cμ−1Tρ, blows up when μ −→0. Since |ψ0| � δ1/2, Tψ0 = T i∂i∂tφ blows up when μ −→ 0 from thedefinition ρ = ψ2
0 .
123
On the formation of shocks for quasilinear wave equations
Corollary 5.2 When shock forms, the only nonzero curvature component inoptical coordinates, αAB, blows up.
Proof In the expression:
αAB = 1
2
d(c2)
dρ/∇2XA,XB
ρ − 1
2μ−1 d(c
2)
dρT (ρ)χ
AB
+1
2
(d2(c2)
dρ2− 1
2c−2
∣∣∣∣d(c2)
dρ
∣∣∣∣
2)
XA(ρ)XB(ρ),
the terms 12d(c2)dρ
/∇2XA,XB
ρ, 12 (d2(c2)dρ2
− 12c
−2|d(c2)dρ |2)XA(ρ)XB(ρ), being deriv-atives of ρ in optical coordinates, are bounded pointwisely. While by (4.7) and
(3.18), the pointwise norm of d(c2)dρ T (ρ)χ
ABis bounded from below, therefore
the term μ−1 d(c2)dρ T (ρ)χ
ABblows up when μ −→ 0. �
The focus of the rest of the paper is to prove Theorem 3.1.
6 Energy estimates for linear wave equations
We study energy estimates for the linear wave equation
�gψ = ρ (6.1)
where ρ is a smooth function defined on Wδ . The energy momentum tensorsfor a solution ψ of (6.1) are the same for both g and g, i.e.,
Tμν = Tμν = ∂μψ∂νψ − 1
2gμν∂
αψ∂αψ.
In the null frame (X1, X2, L , L), Tμν is decomposed as
For the sake of simplicity, we use E(t, u), F(t, u), E(t, u) and F(t, u) asshorthand notations for E(ψ)(t, u), F(ψ)(t, u), E(ψ)(t, u) and F(ψ)(t, u)in the rest of the section.
We need to compute the so called error integrals or error terms, i.e.,∫W
utQ0
and∫W
utQ1 in (6.5). This requires an explicit formula for π0,μν or π1,μν . The
123
On the formation of shocks for quasilinear wave equations
deformation tensor π0,μν is given by
π0,LL = 0, π0,LL = 0, π0,LL
= −2c−1μ(μ−1Lμ + L log(c−1) + L(c−2μ)),
π0,L A = −2c−1μXA(c−2μ), π0,L A
= −2c−1(ζA
+ ηA), π0,AB = 2χAB .
(6.8)
The deformation tensor π1,μν is given by
π1,LL = 0, π1,LL = 4c−1μL(c−2μ),
π1,LL = −2c−1μ(μ−1Lμ + L log(c−1)),
π1,L A = 0, π1,L A = 2c−1(ζA
+ ηA
), π1,AB = 2χ
AB.
(6.9)
We also need an explicit formula for the energy momentum tensor Tμν :
T LL = (Lψ)2
4μ2 , T LL = (Lψ)2
4μ2 ,
T LL = (/dψ)2
4μ, T LA = −LψXA(ψ)
2μ,
T LA = −LψXA(ψ)
2μ,
T AB = /gAC/gBDXC (ψ)XD(ψ) − 1
2/gAB
(−LψLψ
μ+ |/dψ |2
).
(6.10)
Finally, we can compute the integrands Q0 and Q1 explicitly. For Q0, we have
c−2 Q0 = −c−2ρ · K0ψ − 1
2Tμνπ0,μν
= Q0,0 + Q0,1 + Q0,2 + Q0,3 + Q0,4
= −c−2ρ · K0ψ − T LL π0,LL − T LAπ0,L A
− T LAπ0,L A − 1
2T AB π0,AB .
(6.11)
123
S. Miao, P. Yu
The Q0,i ’s are given explicitly by
Q0,1 = 1
2c(μ−1Lμ + L log(c−1) + L(c−2μ))|/dψ |2,
Q0,2 = −c−1XA(c−2μ)Lψ · XA(ψ),
Q0,3 = −c−1μ−1(ζA
+ ηA)LψXA(ψ),
Q0,4 = −1
2
(χ AB X
AψXBψ + c
2μ−1trgχ Lψ · Lψ).
(6.12)
For Q1, we have
c−2 Q1 = −c−2ρ · K1ψ − 1
2Tμνπ1,μν=Q1,0+Q1,1 + Q1,2 + Q1,3 + Q1,4
= −c−2ρ · K1ψ − 1
2T LL π1,LL − T LL π1,LL − T LAπ1,L A
−1
2T AB π1,AB . (6.13)
The Q1,i ’s are given explicitly by
Q1,1 = − 1
2cμL(c−2μ)|Lψ |2, Q1,2 = 1
2c(μ−1Lμ + L log(c−1))|/dψ |2,
Q1,3 = 1
cμ
(ζA
+ ηA
)Lψ · X Aψ,
Q1,4 = −1
2
(χ
ABX AψXBψ + c
2μ−1trgχ Lψ · Lψ).
(6.14)
6.1 Estimates on Q1,2: the coercivity of energy estimates in shock region
We separate the principal terms and lower order terms of Q1,2 as follows
Q1,2 = 1
2c
(μ−1Lμ + L log(c−1)
)|/dψ |2 =(
− 1
2cμ−1Lμ + l.o.t.
)|/dψ |2,
where the lower order terms l.o.t., thanks to (3.10), are bounded by‖l.o.t.‖L∞(�t ) � δM2. We rewrite the principal term
∫W
ut
12cμ
−1Lμ|/dψ |2as
∫
Wut
1
2cμ−1Lμ|/dψ |2 =
∫
Wut ∩Wshock
1
2cμ−1Lμ|/dψ |2
+∫
Wut ∩Wrare
1
2cμ−1Lμ|/dψ |2. (6.15)
123
On the formation of shocks for quasilinear wave equations
In view of the estimates (4.5) and the fact that μ ≥ 110 in the rarefaction wave
region Wrare, for sufficiently small δ, the second term is bounded by
∫
Wut ∩Wrare
1
2cμ−1Lμ|/dψ |2 �
∫
Wut ∩Wrare
|Lμ||/dψ |2
=∫
Wut ∩Wrare
∣∣∣∣|r0|2|t |2 Lμ(−r0, u, θ) + δM2
∣∣∣∣ |/dψ |2
�∫
Wut ∩Wrare
|/dψ |2.
Since∫W
ut ∩Wrare
|/dψ |2 ≤ ∫Wut
|/dψ |2 = ∫ t−r0
( ∫�τ
μ|/dψ |2)dτ , we obtain∫
Wut ∩Wrare
1
2cμ−1Lμ|/dψ |2 �
∫ t
−r0E(τ, u)dτ. (6.16)
In the shock region, we make use of the key estimate (4.7). Therefore, wehave
∫
Wut ∩Wshock
1
2cμ−1Lμ|/dψ |2 ≤ −
∫
Wut ∩Wshock
1
8c|t |μ−1|/dψ |2
� −∫
Wut ∩Wshock
μ−1|/dψ |2.
We define
K (t, u) =∫
Wut ∩Wshock
μ−1|/dψ |2, (6.17)
Therefore,∫
Wut ∩Wshock
1
2cμ−1Lμ|/dψ |2 � −K (t, u). (6.18)
Finally, by combining (6.15), (6.16) and (6.18), for sufficiently small δ, weobtain
∫
Wut
Q1,2 � −K (t, u) +∫ t
−r0E(τ, u)dτ. (6.19)
The negative term −K (t, u) in above estimates plays a key role to control /dψin error terms. It will compensate the degeneracy of the μ factor in front of/dψ in the energy.
123
S. Miao, P. Yu
6.2 Estimates on Q0,1
The estimates on Q0,1 relies on the second key property (4.23) of μ. We firstseparate the principal terms and lower order terms of Q0,1 as follows:
Q0,1 = 1
4c(μ−1Lμ + L log(c−1) + L(c−1κ))|/dψ |2
= 1
2cμ−1Tμ|/dψ |2 + l.o.t.,
where l.o.t. = 14c (c
−2Lμ + L log(c−1) + L(c−1κ))|/dψ |2. According to theestimates derived in previous sections, the terms in the parentheses are boundedby M2. Hence,
∫
Wut
l.o.t. �∫
Wut
M2|/dψ |2 = M2∫ u
0
(∫
Cu′μ|/dψ |2
)
du′
≤ M2∫ u
0F(t, u′)du′. (6.20)
For the principal termwhose integrand is 12cμ
−1Tμ|/dψ |2 ∼ μ−1Tμ|/dψ |2,due to the positivity of |/dψ |2, one ignores the contribution from the negativepart 1
c (μ−1Tμ)−|/dψ |2. Therefore, it is bounded by
�∫
Wut
(μ−1Tμ)+|/dψ |2 =∫
Wut ∩Wrare
(μ−1Tμ)+|/dψ |2
+∫
Wut ∩Wshock
(μ−1Tμ)+|/dψ |2.
In Wrare, since μ−1 � 1, we have∫W
ut ∩Wrare
(μ−1Tμ)+|/dψ |2 �∫W
ut ∩Wrare
|Tμ||/dψ |2 ≤ ∫Wut
|Tμ||/dψ |2. According to (4.15), we have Tμ �δ−1, therefore,∫
Wut ∩Wrare
(μ−1Tμ)+|/dψ |2 �∫
Wut
δ−1|/dψ |2 ≤ δ−1∫ u
0F(t, u′)du′. (6.21)
In Wshock , the argument relies on Proposition 4.16. Indeed, we have∫
Wut ∩Wshock
(μ−1Tμ)+|/dψ |2 �∫
Wut ∩Wshock
1
|t − s∗| 12δ−1|/dψ |2
≤ δ−1∫
Wut
1
|t − t∗| 12|/dψ |2.
123
On the formation of shocks for quasilinear wave equations
Bydefinition, the last integral is equal to δ−1∫ t−r0
1
|τ−t∗| 12( ∫
�τμ|/dψ |2dμ/g
)dτ .
In view of the definition of E(t, u), we obtain
∫
Wut ∩Wshock
(μ−1Tμ)+|/dψ |2 � δ−1∫ t
−r0
1
|τ − t∗| 12E(τ, u)dτ (6.22)
We remark that the key feature of above estimates is that the factor 1
|t−t∗| 12is
integrable in t . It will allow us to use Gronwall’s inequality.Finally, taking into account of the estimates (6.20), (6.21) and (6.22), for
sufficiently small δ, we obtain
∫
Wut
Q0,1 � δ−1∫ t
−r0
1
|τ − t∗| 12E(τ, u)dτ + δ−1
∫ u
0F(t, u′)du′. (6.23)
6.3 Estimates on other error terms
We deal with Q0,2, Q0,3, Q0,4, Q1,1, Q1,3 and Q1,4 one by one.For Q0,2, we have |Q0,2| = |− 1
c XA(c−2μ)Lψ ·XA(ψ)| � |/dμ||Lψ ||/dψ |.
According to (4.13), |/dμ| � 1+δM4. Hence, for sufficiently small δ, we have
∣∣∣∣∣
∫
Wut
Q0,2
∣∣∣∣∣�∫
Wut
(1 + δM4)|Lψ ||/dψ | �∫
Wut
|Lψ |2 + |/dψ |2
Therefore, we obtain
∣∣∣∣∣
∫
Wut
Q0,2
∣∣∣∣∣�∫ t
−r0E(τ, u)dτ. (6.24)
For Q0,3, we first recall that |ζ | � 1 and η = ζ + /dμ, therefore, |ζ |+ |η| �1 + δM4. We break Q0,3 into two parts as follows:
∣∣∣∣∣
∫
Wut
Q0,3
∣∣∣∣∣�∫
Wut
μ−1(|ζ | + |η|)|Lψ ||/dψ | �∫
Wut
μ−1|Lψ ||/dψ | = I1 + I2
=∫
Wut ∩Wrare
μ−1|Lψ ||/dψ | +∫
Wut ∩Wshock
μ−1|Lψ ||/dψ |.
123
S. Miao, P. Yu
The integral I1 = ∫W
ut ∩Wrare
μ−1|Lψ ||/dψ | is taken in Wrare where μ ≥ 110 ,
therefore, we obtain
I1 ≤∫
Wut ∩Wrare
μ−1(|Lψ |2 + |/dψ |2) �∫
Wut ∩Wrare
μ−1(|Lψ |2 + μ|/dψ |2)
�∫
Wut
μ−1(|Lψ |2 + μ|/dψ |2)
�∫ t
−r0E(τ, u)dτ +
∫ u
0F(t, u′)du′.
To control the integral I2 = ∫W
ut ∩Wshock
μ−1|Lψ ||/dψ |, we use the coercive
term K (t, u) from Q1,2 to control the loss of μ−1:
I2 ≤(∫
Wut ∩Wshock
μ−1|/dψ |2) 1
2(∫
Wut ∩Wshock
μ−1|Lψ |2) 1
2
� K (t, u)12
(∫ t
−r0E(τ, u)dτ
) 12
.
Hence, we obtain∣∣∣∣∣
∫
Wut
Q0,3
∣∣∣∣∣�∫ t
−r0E(τ, u)dτ +
∫ u
0F(t, u′)du′
+ K (t, u)12
(∫ t
−r0E(τ, u)dτ
) 12
. (6.25)
For Q0,4, we have∣∣∣∣∣
∫
Wut
Q0,4
∣∣∣∣∣�∫
Wut
|χ ||/dψ |2 + μ−1|trgχ ||Lψ ||Lψ |
�∫
Wut
δM2|/dψ |2 + μ−1|Lψ ||Lψ | = I1 + I2.
The bound on I1 is immediate: I1 = δM2∫W
ut
|/dψ |2 ≤ δM2∫ u0 F(t, u′)du′.
The bound on I2 = ∫Wutμ−1|Lψ ||Lψ | relies on the energy E(t, u) and on the
flux F(t, u):
I2 �∫
Wut
μ−1|Lψ |2 +∫
Wut
μ−1|Lψ |2 =∫ t
−r0E(τ, u)dτ +
∫ u
0F(t, u′)du′.
123
On the formation of shocks for quasilinear wave equations
Hence, we obtain
∣∣∣∣∣
∫
Wut
Q0,4
∣∣∣∣∣�∫ t
−r0E(τ, u)dτ + δM2
∫ u
0F(t, u′)du′ +
∫ u
0F(t, u′)du′.
(6.26)
For Q1,1, according to (4.5), we have∣∣Q1,1
∣∣ = ∣∣ 12cμL(c
−2μ)|Lψ |2∣∣ �∣∣μ−1Lμ|Lψ |2∣∣ � μ−1|Lψ |2. Hence,
∣∣∣∣∣
∫
Wut
Q1,1
∣∣∣∣∣�∫ u
0F(t, u′)du′. (6.27)
For Q1,3, since |ζ | + |η| � 1, we break the integral into two parts:
∣∣∣∣∣
∫
Wut
Q1,3
∣∣∣∣∣�∫
Wut
μ−1|Lψ ||/dψ | = I1 + I2
=∫
Wut ∩Wrare
μ−1|Lψ ||/dψ | +∫
Wut ∩Wshock
μ−1|Lψ ||/dψ |.
In the rarefaction wave region, since μ−1 ∼ 1, we obtain
I1 �∫
Wut ∩Wrare
μ−1(|Lψ |2 + |/dψ |2) ≈∫
Wut ∩Wrare
|Lψ |2 + |/dψ |2
�∫
Wut
|Lψ |2 + |/dψ |2 �∫ t
−r0E(τ, u)dτ.
In the shock wave region, we still use K (t, u) to control the loss of μ−1:
I2 ≤(∫
Wut ∩Wshock
μ−1|/dψ |2) 1
2(∫
Wut ∩Wshock
μ−1|Lψ |2) 1
2
� K (t, u)12
(∫ u
0F(t, u′)du′
) 12
.
Hence,
∣∣∣∣∣
∫
Wut
Q1,3
∣∣∣∣∣�∫ t
−r0E(τ, u)dτ + K (t, u)
12
(∫ u
0F(t, u′)du′
) 12
. (6.28)
123
S. Miao, P. Yu
For Q1,4, we have
∣∣∣∣∣
∫
Wut
Q1,4
∣∣∣∣∣�∫
Wut
|χ ||/dψ |2 + μ−1|trgχ ||Lψ ||Lψ |
�∫
Wut
δM2|/dψ |2 + μ−1|Lψ ||Lψ |
� δM2∫ u
0F(t, u′)du′
+(
δ
∫
Wut
μ−1|Lψ |2 + δ−1∫
Wut
μ−1|Lψ |2)
.
Therefore, we obtain
∣∣∣∣∣
∫
Wut
Q1,4
∣∣∣∣∣� δ
∫ t
−r0E(τ, u)dτ + δM2
∫ u
0F(t, u′)du′
+ δ−1∫ u
0F(t, u′)du′. (6.29)
6.4 Summary
In view of (6.5) and (6.7), since ψ vanishes to infinite order on C0, we have
Remark 6.1 We see that the energy estimates only imply
E(t, u) + K (t, u) � δ,
which does not recover the full regularity of Lψ and /d psi with respect to δ, asindicated by bootstrap assumption. We will see finally that this full regularityin δ is recovered by using the estimates for E(t, u) together with Lemma 7.3and the commutation of Q.
7 Comparisons between Euclidean and optical geometries
There are two different geometries coming into play on W ∗δ , namely, the
Minkowski geometry and the optical geometry. The knowledge on twometricsgμν (or gμν) andmμν is essentially tied to the estimates on the solution of (�).There are many ways to compare two geometries, e.g., we may consider theCartesian coordinates xk as functions of the optical coordinates (t, u, θ). Inwhat follows, we also study other quantities as yk , zk , λi , etc. As a by prod-uct, we will also obtain estimates for the lower order objects, i.e., with order<Ntop + 1.
Given a vectorfield V , we define the null components of its deformationtensor as
(V )Z A= (V )π(L, XA),(V )ZA = (V )π(L , XA),
(V ) /π AB = (V )π(XA, XB).
(7.1)
The projection of Lie derivative LV to St,u is denoted as /LV . The short-hand notation /Lα
Zi to denote /LZi1/LZi2
· · · /LZikfor a multi-index α =
(i1, . . . , ik). We will show that, for all |α| ≤ N∞, we have /Lα−1Zi
χ ′, /Lα−1Zi
(Z j )Z , /Lα−1Zi
(Z j ) /π, /Lα−1Zi
(Q) (/π + 4) ∈ O|α|2−2l , where l is the num-
ber of T ’s in Zi ’s, |α| ≥ 1 and Z j /=T . If Z j = T , then we have/Lα−1Zi
(T )Z , /Lα−1Zi
(T ) /π ∈ O|α|−2l . The idea of the proof is to compare gμν and
mμν via quantities such as xi , yi , zi , T i and λi . Similarly, we will deriveL2-estimates on objects of order ≤ Nμ. The L2 estimates depend on the L∞
123
On the formation of shocks for quasilinear wave equations
estimates up to order N∞ + 2. In the course of the proof, it will be clear whyN∞ is chosen to be approximately 1
2Ntop.
7.1 L∞ estimates
We assume (B.1): for all |α| ≤ N∞, Zα+2i ψ ∈ �
|α|+21−2l .
Proposition 7.1 For sufficiently small δ, for all |α| ≤ N∞ and t ∈ [−r0, s∗],we can bound
{/LαZiχ
′, /LαZi(Z j )Z , /Lα
Zi(Z j ) /π, /Lα
Zi(Q) (/π + 4) , Z α+1
i y j ,
Z α+1i λ j } ⊂ O|α|+1
2−2l and {Z α+1i x j , Z α+1
i T j , /Lα+1Zi Z j , /Lα
Zi(T )Z , /Lα
Zi(T ) /π} ⊂
O|α|+1−2l in terms of Zα+2
i ψ ∈ �|α|+21−2l . Here l is the number of T ’s in Zi ’s.
The estimates on x j should be viewed as a good comparison betweenEuclid-ean and optical geometries on each slice �t .
Proof We do induction on the order. When |α| = 0, the estimates are treatedin Sect. 3. Here we only treat the estimates when l = 0, when l ≥ 1, we canuse the structure equation (2.34) to reduce the problem to the estimates for μ,which will be treated in Proposition 7.2. Here the loss of δ in the estimatesfor /Lα
Zi(T )Z , /Lα
Zi(T ) /π comes from applying T to 2
t−u , which is the principalpart of trχ . Given |α| ≤ N∞, we assume that estimates hold for terms of order
≤ |α|. In particular, we have /LβRiχ ′, R β+1
i y j ∈ O|β|2 for all |β| ≤ |α|. We
prove the proposition for |α| + 1.Step 1 Bounds R α+1
i x j . Let δ jα+1,i = α+1i x j−R α+1
i x j where i ’s are the
standard rotational vectorfields on Euclidean space. It is obvious that α+1i x j
is equal to some xk , therefore, bounded by r hence by a universal constant.Since Ri = i − λi T j∂ j , Ri x j ∈ O0
0 and by ignoring all the numericalconstants, we have
δjα+1,i = Rα
i
(λi T
j)
= Rαi
(λi
(x j
u − t+ y j
)).
Here the index i is not a single index. It means we apply a string of differentRi s. This notation applies in the following when a string of Ri s are considered.Since the above expression has total order≤|α|, by the induction hypothesis
we obtain immediately that δ jα+1,i ∈ O|α|2 . In view of the definition of δ jα+1,i ,
we then have R α+1i x j ∈ O|α|+1
0 .
123
S. Miao, P. Yu
Step 2 Bounds on /LαRiχ
′. We commute /LαRi with (3.19) to derive
/LL /LαRiχ
′ = [/LL , /LαRi
]χ ′ + e · /Lα
Riχ′ + χ ′ · /Lα
Riχ′
+∑
|β1|+|β2|=|α||β2|<|α|
Riβ1e · /Lβ2
Riχ ′
+∑
|β1|+|β2|+|β3|=|α||β1|<|α|,|β3|<|α|
/Lβ1Riχ ′ · /Lβ2
Ri /g−1 · /Lβ3
Riχ ′
+/LαRi
(e/gAB
t − u− α′
AB
). (7.2)
Since e = c−1 dcdρ Lρ, Ri
αe is of order ≤ |α| + 1. By (B.1), we have Rαi e ∈
�|α|+12 . Similarly, by the explicit formula of α′
AB , we have /LβRiα′AB ∈ �
|α|+22 .
Since (Ri )πAB = 2c−1λiχ ABand /Lβ2
Ri /g = /Lβ2−1
Ri(Ri ) /π AB , by the estimates
derived in previous sections and by the induction hypothesis, we can rewrite(7.2) as
/LL /LαRiχ
′ = [/LL , /LαRi
]χ ′ + O1
2 · /Lα−1Ri χ ′ + �
≤|α|+22 .
The commutator can be computed as[/LL , /Lα
Ri
]χ ′ = ∑
|β1|+|β2|=|α|−1
/Lβ1Ri/L(Ri )Z
/Lβ2Riχ ′. Since (Ri )Z A = −χ ′
ABRi
B + εi jk z j X Ak + λi /d A(c) and
z j = − (c−1)x j
u−t − cy j , the commutator term is of type O12 · /Lα
Riχ′. Therefore,
we have
/LL /LαRiχ
′ = O12 · /Lα
Riχ′ + �
≤|α|+22 .
By integrating this equation from −r0 to t , the Gronwall’s inequality yields‖/Lα
Riχ′‖L∞(�t ) �M δ.
Step 3 Bounds on R α+1i y j and R α+1
i λ j . Since Ri y j = (−c−1χ AB
−δABu−t )R
Ai/dBx j , schematically we have
R α+1i y j = Rα
i Rk yj = Rα
i
((c−1χ + δ
u − t
)· Rk · /dx j
).
We distribute Rαi inside the parenthesis by Leibniz rule. (Here again, the index
i is not a single index, so we use index k to distinct the last rotation vectorfield.)Therefore, a typical termwould be either /LRi
β1χ · /LRiβ2/g−1 · /LRi
β3Rk ·/dRβ4i x j
123
On the formation of shocks for quasilinear wave equations
or /LRiβ1/g · /LRi
β2/g−1 · /LRi
β3Rk · /dRβ4i x j with |β1|+ |β2|+ |β3|+ |β4| = |α|.
There are only two terms where are not included in the induction hypoth-esis: /LRi
β2/gAB and /LRi
β3R j . The first term is in fact easy to handle byinduction hypothesis and estimates derived in Step 1 and Step 2, since/LRi /gAB = (Ri ) /π AB = 2λi c−1χ
AB. For the second one, we use the following
expression:
/LRi R j = −3∑
k=1
εi jk Rk + λi
(c−1 − 1
u − t/g − c−1
(χ + /g
u − t
))R j
− λ j
(c−1 − 1
u − t/g − c−1
(χ + /g
u − t
))Ri
− λiε jkl yk /dxl · /g−1 + λ jεikl y
k /dxl · /g−1.
Therefore, /LRiβ3Rk = /LRi
β3−1 /LRi Rk = O|β3|−10 + O|β3|−1
≥2 Rβ3i x j . Finally,
we obtain that
R α+1i y j = O≤|α|
2 + O≤|α|2 · /dRβ4
i x j .
Although /dRβ4i x j and /LRi
β1χ may have order |α| + 1, they have been con-
trolled from previous steps. This gives the bounds on R α+1i y j . Then by the
fact that λ j = ε jkl xk yl , the estimate for R α+1i λ j follows. This completes the
proof of the proposition. � Proposition 7.2 For sufficiently small δ, for all |α| ≤ N∞, t ∈ [−r0, s∗], wecan bound Zα+1
i μ ∈ O|α|+1−2l in terms of Zα+2
i ψ ∈ �|α|+21−2l .
Proof We use an induction argument on the order of derivatives. The base case|α| = 0 has be treated in Sects. 3 and 4. We assume the proposition holds withorder of derivatives on μ at most |α|. For |α| + 1, by commuting Zα+1
i withLμ = m + μe, we have
Lδl Zα+1i μ=(e+(Zi )Z)δl Zα+1
i μ + δl Zα+1i m +
∑
|β1|+|β2|≤|α|+1|β1|<|α|
δl1 Zβ1i μδl2 Zβ2
i e
where la, a = 1, 2 is the number of T ’s in Zβa ’s and l is the number of T ’s inZα’s. By the induction hypothesis, the above equation can be written as:
Lδl Zα+1i μ = O≤1
2 · δl Zα+1i μ + �
≤|α|+20
123
S. Miao, P. Yu
Similar to the estimates derived in the Step 3 in previous section, wecan use induction hypothesis and Gronwall’s inequality to conclude that‖Zα+1
i μ‖L∞(�t ) �M δ−l . �
7.2 L2 estimates
Ntop will be the total number of derivatives commuted with �gψ = 0. Thehighest order objects will be of order Ntop + 1. In this subsection, based on(B.1), we will derive L2 estimates on the objects of order ≤ Ntop in terms of
the L2 norms of Zα+2i ψ ∈ �
|α|+21−2l with |α| ≤ Ntop − 1. We start with the
following lemma:
Lemma 7.3 For a smooth function ψ which vanishes on C0, we have
∫
St,uψ2 � δ
∫
�ut
(Lψ)2 + μ(Lψ)2,∫
�ut
ψ2 � δ2∫
�ut
(Lψ)2 + μ(Lψ)2.
(7.3)
Proof Since ψ(t, u, θ) = ∫ u0 Tψ(t, u′, θ)du′, we have
∫
St,uψ2dμ/g =
∫
St,u
(∫ u
0Tψ(t, u′, θ)du′
)2
dμ/g(t,u)
� δ
∫
St,u
∫ u
0(Tψ(t, u′, θ))2du′dμ/g(t,u)
� δ
∫
St,u
∫ u
0(Tψ(t, u′, θ))2dμ/g(t,u′)du
′
Here we have used the fact:√det /g(t, u) �
√det /g(t, u′) �
√det /g(t, u) due
to the bound of the second fundamental form θ . On the other hand, (Tψ) �(Lψ)2 + μ2(Lψ)2 and μ � 1, the first inequality follows immediately. Thesecond is an immediate consequence of the first one. �
As a corollary, for k ≤ Ntop − 1, we have
∑
|β|≤k
∫
St,u(Ri
βψ)2 � δE≤k+1(t, u). (7.4)
Proposition 7.4 For sufficiently small δ, for all α with |α| ≤ Ntop − 1 andt ∈ [−r0, s∗], the L2(�
ut ) norms of all the quantities listed below
/LαZiχ
′, /LαZi(Z j )Z , /Lα
Zi(Z j ) /π, Z α+1
i y j , Z α+1i λ j ,
123
On the formation of shocks for quasilinear wave equations
are bounded6 by δ1/2−l∫ t−r0
μ−1/2m (t ′)
√E≤|α|+2(t
′, u)dt ′,where l is the num-ber of T ’s in Zi ’s.
Proof We do induction on the order. When |α| = 0, the result follows fromthe estimates in Sects. 3 and 4. Again, here we only treat the case l = 0. Byassuming the proposition holds for terms with order ≤ |α|, we show it holdsfor |α| + 1.
Step 1 Bounds on /LαRiχ
′. By affording a L-derivative, we have
‖/LαRiχ
′‖L2(�ut )
� ‖/LαRiχ
′‖L2(�u−r0
) +∫ t
−r0‖|χ ′||/Lα
Riχ′|
+ |/LL /LαRiχ
′|‖L2(�uτ )dτ. (7.5)
We use formula (7.2) to replace /LL /LαRiχ
′ by the terms with lower orders. Eachnonlinear term has at most one factor with order >N∞. We bound this factorin L2(�t ) and the rest in L∞. We now indicate briefly how the estimates onthe factors involving e and α′ work.
For α′, since
α′AB = c
dc
dρ/D2A,Bρ + 1
2
[d2(c2)
dρ2− 1
2c2
(dc2
dρ
)2]
XA(ρ)XB(ρ),
in view of the definition of E(t, u), for sufficiently small δ, we have
‖/LαRiα
′AB‖L2(�
ut )
�∑
|α|≤k
‖/dRα+1i ψ‖L2(�
ut )
�M δ12μ
−1/2m (t)
√E≤|α|+2(t, u).
For e, since e = c−1 dcdρ Lρ, we have
‖Rαi e‖L2(�
ut )
�M
∑
|β|≤|α|
(δ1/2‖/dR|β|−1
i ψ‖L2(�ut )
+δ1/2‖/dR|β|−1i Lψ‖L2(�
ut )
)
�M δ1/2μ−1/2m (t)
√E≤|α|+1(t, u).
By applying Gronwall’s inequality to (7.5), we obtain immediately that
‖/LαRiχ
′‖L2(�ut )
�M δ1/2∫ t
−r0μ
−1/2m (τ )
√E≤|α|+2(τ, u)dτ
6 The inequality is up to a constant depending only on the bootstrap constant M .
123
S. Miao, P. Yu
Step 2 Bounds on R α+1i y j . By the computations in the Step 2 of the proof
of Proposition 7.1, R α+1i y is a linear combination of the terms such as /LRi
β1χ ·/LRi
β2/g−1·/LRi
β3R j ·/dRβ4i x j or as /LRi
β1/g·/LRi
β2/g−1·/LRi
β3R j ·/dRβ4i x j , where
|β1|+|β2|+|β3|+|β4| = |α|. Similarly, we bound all factors with order≤ N∞by the L∞ estimates in Proposition 7.1. By the induction hypothesis, this yieldsthe bound on R α+1
i y j immediately. The estimates for other quantities follow
from the estimates of χ ′ and y j . In this process, the terms like Rβi x
j and theleading term in /LRi R j , which can be bounded by a constantC disregarding theorder of the derivatives, are bounded in L∞. The rest terms in /LRi R j , whichdepend on χ ′ and y j as well as their derivatives, are bounded in L2 based on
the L2 estimates for /LβRiχ ′. �
We also have L2 estimates for derivatives of μ.
Proposition 7.5 For sufficiently small δ, for all α with |α| ≤ Ntop − 1 andt ∈ [−r0, s∗], we have
δl‖Zα+1i μ‖L2(�t )
�M δl‖Zα+1i μ‖L2(�
u−r0
) + δ1/2∫ t
−r0
√E≤|α|+2(τ, u)
+μ−1/2m (τ )
√E≤|α|+2(τ, u)dτ.
Proof According to the proof of Proposition 7.2, we have
δl L|Zα+1i μ| � δl |Zα+1
i m| + δl(|e| + |(Ri )Z |)|Zα+1i μ|
+∑
|β1+β2|≤|α|δl1 |Zβ1
i μ|δl2 |Rβ2i e|.
Then the result follows in the same way as Proposition 7.2. �
8 Estimates on top order terms
The highest possible order of an object in the paper will be Ntop + 1. Thecurrent section is devoted to the L2 estimates of /dR α
i trχ and Z α+2i μ with
|α| = Ntop − 1.
8.1 Estimates on trχ
We first sketch the idea of the proof. Since we deal with top order terms, wecan not use the transport equation (2.35) directly as in the previous section
123
On the formation of shocks for quasilinear wave equations
(which loses one derivative). Roughly speaking, we derive an elliptic systemcoupled with a transport equation for χ and /dtrχ :
L(/dRαi trχ) = χ · /∇Lα
Ri χ + · · · , /divLαRi χ = /dRα
i trχ + · · · .
The new idea is using elliptic estimates and rewriting the right hand side ofthe transport equations to avoid the loss of derivatives.
Given �gψ0 = 0, since ρ = ψ20 , we can derive a wave equation for ρ:
�gρ = d(log(c))
dρgμν∂μρ∂νρ + 2gμν∂μψ0∂νψ0 (8.1)
Therefore, in the null frame, we can rewrite �/ρ as /�ρ = μ−1L(Lρ) + l.o.t.
where l.o.t. represents all the terms with order at most 1. On the otherhand, according to the definition of αAB , we can rewrite (2.35) as Ltrχ =−1
2d(c2)dρ
/�ρ + l.o.t. where the lower order terms l.o.t. standard for terms withorder at most 1. By substituting to the previous expression on /�ρ, we obtain:
L(μtrχ − f
) = 2Lμtrχ − 1
2μ(trχ)2 − μ|χ |2 + g, (8.2)
where f = −12d(c2)dρ Lρ and g is given by
g =(
2
(d(c)
dρ
)2
+ cd2(c)
dρ2
)(LρLρ − μ|/dρ|2)
+ 2cd(c)
dρ
((Lψ0Lψ0 − μ|/dψ0|2
)+(1
4
μ|/dρ|2c2
− ζ A/d Aρ
)).
We observe twomain features of (8.2): the order of the righthand side termsare one less than that of the lefthand side; It is regular in μ, i.e. there is no μ−1
factor. In order to commute Ri ’s with (8.2) and to control the αth derivativesof /d(trχ), for a given multi-index α, we introduce
Fα = μ/d(Riαtrχ) − /d(Ri
α f ).
For α = 0 and F = F0 = μ/dtrχ − /d f , by commuting /d with (8.2), weobtain
/LL F + (trχ − 2μ−1Lμ)F =(
−1
2trχ + 2μ−1Lμ
)/d f − μ/d(|χ |2) + g0
123
S. Miao, P. Yu
with g0 = /dg− 12 trχ /d( f −2Lμ)−(/dμ)(Ltrχ+|χ |2). Similarly, for |α| �= 0,
we first commute R αi and then commute /d with (8.2). This leads to
/LL Fα + (trχ − 2μ−1Lμ)Fα =(
−1
2trχ + 2μ−1Lμ
)/d(Ri
α f )
−μ/d(Riα(|χ |2)) + gα, (8.3)
with gα in the following schematic expression (by setting all the numericalconstants to be 1):
gα = /LRiαg0 +
∑
|β1|+|β2|=|α|−1
/Lβ1Ri/L(Ri )Z Fβ2
+∑
|β1|+|β2|=|α|−1
/Lβ1Ri
((μRi trχ + Ri Lμ + (Ri )Zμ
)/d(Ri
β2 trχ))
+∑
|β1|+|β2|=|α|−1
/Lβ1Ri
(Riμ
[/LL /d
(Ri
β2 trχ)+ trχ/d
(Ri
β2 trχ)
+ /d(Ri
β2(|χ |2))]+ Ri trχ/d(Ri
β2 f)).
We remark that Fα is of order Ntop + 1 so that α = Ntop − 1.We rewrite (2.33) as /divχ = 1
2/dtrχ − (μ−1ζ · χ − 1
2μ−1ζ trχ). By com-
muting /LRiα , we obtain the following schematic expression:
(Ri ) /π)β1((Ri ) /π ·∇/ /LRiβ2 χ ). By applying the elliptic estimates (3.44) to (8.4),
in view of the definition of Fα , we obtain (the L2 norms are taken at S(t, u))
‖μ/∇ /LRiαχ‖L2(St,u) � ‖Fα‖L2 + ‖/dRi
α f ‖L2 + ‖/dμ‖L∞‖ /LRiαχ‖L2
+ ‖μHα‖L2 . (8.5)
For any form ξ , since |ξ |L|ξ | = (ξ, /LLξ) − ξ · χ · ξ − 12 trχ |ξ |2, we have
L|ξ | ≤ |/LLξ |+ |χ ||ξ |− 12 trχ |ξ |. Applying this inequality to (8.3), we obtain
L|Fα| ≤(μ−1Lμ − 3
2trχ + |χ |
)|Fα| + (2μ−1|Lμ| − trχ)|/dRi
α f |+ |μ/dRi
α(|χ |2)| + |gα|. (8.6)
123
On the formation of shocks for quasilinear wave equations
To obtain the L2(�t ) bound on Fα , we first integrate (8.6) from −r0 to t andthen take the L2 norm on�t . We claim that we can ignore the first term on theright hand side: The term
(|trχ | + |χ |)|Fα| can be removed immediately by
Gronwall’s inequality. Forμ−1Lμ · |Fα|, for a fixed (u, θ), ifμ ≥ 110 for all u,
it can be also removed by Gronwall’s inequality; otherwise,μ < 110 , therefore,
according to the argument in Proposition 4.6, the sign of Lμ is negative sothat this term can be ignored. As a result, (8.6) yields
‖Fα‖L2(�t )� ‖Fα‖L2(�−r0 )
+∫ t
−r0‖(μ−1|Lμ| + 1)/dRi
α f ‖L2(�τ )
+‖μ/dRiα(|χ |2)‖L2(�τ )
+ ‖gα‖L2(�τ )dτ
= ‖Fα‖L2(�−r0 )+ I1 + I2 + I3.
where the Ii ’s are defined in the obvious way.We first bound I2. According to the Leibniz rule, we have
I2 =∑
|β1|+|β2|+|β3|+|β4|=|α|∫ t
−r0‖μ/d( /LRi
β1/g · /LRiβ2/g · /LRi
β3 χ · /LRiβ4 χ)‖L2(�τ )
dτ.
Therefore, at least three indices of the βi ’s are at most N∞. According toProposition 7.4, we have
I2 �∫ t−r0
‖μ/d /LRiαχ‖L2(�τ )
‖χ‖L∞(�τ ) + δ3/2√E≤|α|+2(τ, u)dτ
(8.5)� δ
∫ t−r0
‖Fα‖L2(�τ )+ ‖/dRi
α f ‖L2(�τ )+ δ1/2
√E≤|α|+2(τ, u)dτ.
Since f =−12dc2dρ Lρ, we have /dRi
α f = dc2dρ ψ0/dRi
αLψ0+/d(∑
β1+β2=α,|β1|≥1
Riβ1
(dc2dρ ψ0
)Ri
β2(Lψ0
)). This leads to
∫ t
−r0‖/dRi
α f ‖L2(�τ )�∫ t−r0
δ12√E≤|α|+2(τ, u)dτ.
Therefore, we have the following bound for I2:
I2 �∫ t
−r0δ‖Fα‖L2(�τ )
+ δ3/2√E≤|α|+2(τ, u) + δ3/2
√E≤|α|+2(τ, u)dτ.
(8.7)
123
S. Miao, P. Yu
We remark that, as long as the terms under consideration are not top orderterms, i.e. not a term of order Ntop + 1, we can simply use the estimates fromprevious section to get the estimates. The reason is as follows: each term isa product of O≤l
k with l ≤ |α|. Only one of the factor is of order l > N∞.We can bound the rest in L∞ and the highest order one in L2, thanks to theestimates derived in the previous section.
To bound I3, as we pointed out above, we only have to take care ofthe terms appearing in gα whose orders are possibly Ntop + 1. The rest ofthem can be easily bounded in L2 by a universal constant times the sum of∫ t−r0
δ12√E≤|α|+2(τ, u) + δ1/2μ
−1/2m (t)
√E≤|α|+2(τ, u)dτ .
We now investigate the possible highest order terms in gα . There are threepossibilities: the first one are the terms of the form
∑|β1|+|β2|=|α|−1 /Lβ1
Ri/L(Ri )Z
Fβ2 . They can be bounded by |(Ri )Z ||Fα| � δ1/2|Fα| provided that δ is suitablysmall. The second possibility is from the (Ltrχ + |χ |2) term of g0. However,Eq. (2.35) says that Ltrχ + |χ |2 = e − trα′, so although it is of the highestorder, the highest order part consists only ∇/ derivatives of ψ (thanks to theexpression of α′), hence can be bounded in the same way as lower order terms.The last possibility is from the term Lψ0Lψ0 appearing in g. They are of toporders and they can not be converted into terms involving only ∇/ derivatives.These terms contribute to gα the terms of the form O≤1
0 · Lψ0 · LRα+1i ψ0 in
gα . The idea is to use the flux to bound this term:
∫ t
−r0‖Lψ0LR
α+1i ψ0‖L2(�
uτ )dτ
� δ−1/2
(∫ t
−r0
∫ u
0
∫
Sτ,u′(LRα′+1
i ψ)2dμ/gdu′dτ)1/2
� δ−1/2(∫ u
0F≤|α|+2(t, u
′)du′)1/2
Therefore, we have the following estimates for I3:
I3 �∫ t
−r0δ12√E≤|α|+2(τ ) + δ1/2μ
−1/2m (τ )
√E≤|α|+2(t)dτ
+ δ−1/2(∫ u
0F≤|α|+2(t, u
′)du′)1/2
(8.8)
We now study the estimates on I1. When μ ≥ 110 , the estimates are straight-
forward. To study the case when μ ≤ 110 , we introduce a few notations (where
a is a positive constant):
123
On the formation of shocks for quasilinear wave equations
t0 = inf
{τ ∈ [−2, t∗)
∣∣μm(t) <
1
10
},
M(t) = max(u,θ),
(t,u,θ)∈Wshock
∣∣(L(logμ))−(t, u, θ)
∣∣,
Ia(t) =∫ t
t0μ−am (τ )M(τ )dτ.
Lemma 8.1 We assume that a is at least 4.
1. For sufficiently large a and for all t ∈ [t0, t∗), we have
Ia(t) � a−1μ−am (t). (8.9)
1′. For sufficiently large a and for all t ∈ [t0, t∗), we have∫ t
t0μ−a−1m (t ′)dt ′ � 1
aμ−am (t)
2. For a ≥ 4 and sufficiently small δ, there is a constant C0 independent of aand δ, so that for all τ ∈ [−r0, t], we have
μam(t) ≤ C0μ
am(τ ) (8.10)
Proof (1) By Proposition 4.6, for t ≥ t0, the minimum of r20 (Lμ)(−r0, u, θ)on [0, δ] × S
2 is negative and we denote it by
− ηm = min(u,θ)∈[0,δ]×S2
{r20 (Lμ)(−r0, u, θ)}. (8.11)
We notice that 1 ≤ ηm ≤ Cm where Cm is a constant depending on the initialdata. In view of the asymptotic expansion for (Lμ)(t, u, θ) in Lemma 4.4, wehave
μ(t, u, θ) = 1 −(1
t+ 1
r0
)r20 (Lμ)(−r0, u, θ) + O
(δM4)
(1
t2− 1
r20
)
.
(8.12)
We fix an s ∈ (t0, t∗) in such a way that t0 ≤ t < s < t∗. There exists(us, θs) ∈ [0, δ] × S
Indeed, one can apply (8.12) to μ(s, um, θm) and μ(s, us, θs) to derive
μ(s, us, θs) = 1 −(1
s+ 1
r0
)(−ηm + dms) + O(δM4)
(1
t2− 1
r20
)
μ(s, um, θm) = 1 −(1
s+ 1
r0
)(−ηm) + O(δM4)
(1
t2− 1
r20
)
,
(8.15)
where the quantity dms > 0 is defined as
dms := ηm + r20 (Lμ)(−r0, us, θs). (8.16)
Since μ(s, us, θs) ≤ μ(s, um, θm), we have
0 < −(1
s+ 1
r0
)dms ≤ O(δM4)
(1
t2− 1
r20
)
.
Hence,
dms ≤ O(δM4). (8.17)
The constants in the above inequalities depend on t0 therefore on ηm and theyare absolute constants. With this preparation, one can derive precise upper andlower bounds for μm(t).
We pick up a (u′m, θ
′m) ∈ [0, δ] × S
2 in such a way that μ(t, u′m, θ
′m) =
μm(t). For the lower bound, by virtue of Lemma 4.4, we have
μm(t) = μ(t, u′m, θ
′m) = μ(s, u′
m, θ′m) +
∫ t
s(Lμ)(t ′, u′
m, θ′m)dt
′
≥ μm(s) +∫ t
s
ηm
−t ′2+ O(δM4)
(−t ′)3dt ′
≥ μm(s) +(ηm − 1
2a
)(1
t− 1
s
). (8.18)
In the last step, we take sufficiently small δ so that O(δM4) ≤ 12a .
123
On the formation of shocks for quasilinear wave equations
For the upper bound, in view of Lemma 4.4 and (8.17), we have
μm(t) ≤ μ(t, us, θs) = μm(s) +∫ t
s(Lμ)(t ′, us, θs)dt ′
= μm(s) +∫ t
s
ηm − dms
−t ′2+ O(δM4)
(−t ′)3dt ′
≤ μm(s) +∫ t
s
ηm
−t ′2+ O(δM4)
t ′2dt ′
≤ μm(s) +(ηm + 1
2a
)(1
t− 1
s
). (8.19)
In the last step, we also take sufficiently small δ so that O(δM4) ≤ 12a .
For Ia(t), first of all, we have
Ia(t) �∫ t
t0
(μm(s) +
(ηm − 1
2a
)(1
t− 1
s
))−a−1
t ′−2dt ′
=∫ τ0
τ
(μm(s) +
(ηm − 1
2a
)(τ − τs)
)−a−1
dτ ′
≤ 1
ηm − 12a
1
a
(μm(s) +
(ηm − 1
2a
)(τ − τs)
)−a
.
Hence,
Ia(t) � 1
a
(μm(s) +
(ηm − 1
2a
)(1
t− 1
s
))−a
≤ 1
a
(μm(s) + (ηm − 1
2a
) (1t − 1
s
))−a
(μm(s) + (ηm + 1
2a
) (1t − 1
s
))−aμ−am (t)
≤ 1
a
((ηm − 1
2a
) (1t − 1
s
))−a
((ηm + 1
2a
) (1t − 1
s
))−aμ−am (t). (8.20)
Since as a → ∞, one has
(ηm − 1
2a
)−a
(ηm + 1
2a
)−a → e1ηm .
The limit is an absolute constant. Therefore, (8.20) yields the proof for part(1) of the lemma. The proof for part (1’) is exactly the same.
123
S. Miao, P. Yu
(2) We start with an easy observation: if μ(t, u, θ) ≤ 1 − 1a , then
Lμ(t, u, θ) � −a−1. In fact, we claim that (1t + 1r0)r20 Lμ(−r0, u, θ) ≥ 1
2a−1.
Otherwise, for sufficiently small δ (say δ1/4 ≤ a−1), according to the expan-sion forμ(t, u, θ), i.e.μ(t, u, θ) = μ(−r0, u, θ)−(1t + 1
r0)r20 Lμ(−r0, u, θ)+
O(δ), we have
μ(t, u, θ) > 1 − 1
2a− Cδ ≥ 1 − 1
a.
which is a contradiction. Therefore Lemma 4.4 implies Lμ(t, u, θ) � −a−1.In particular, this observation implies that, if there is a t ′ ∈ [−r0, s∗], so thatμm(t ′) ≤ 1 − a−1, then for all t ≥ t ′, we have μm(t) ≤ 1 − a−1. Thisallows us to define a time t1, such that it is the minimum of all such t ′ withμm(t ′) ≤ 1 − a−1.
We now prove the lemma. If τ ≤ t1, since μm(t) ≤ 2, we have
μ−am (τ ) ≤
(1 − 1
a
)−a
≤ C0 ≤ C0μ−am (t).
If τ ≥ t1, then μm(τ ) ≤ 1 − 1a . Let μm(τ ) = μ(τ, uτ , θτ ). We know that
μ(t, uτ , θτ ) is decreasing in t for t ≥ τ . Therefore, we have
μm(t) ≤ μ(t, uτ , θτ ) ≤ μ(τ, uτ , θτ ) = μm(τ ).
The proof now is complete. � For I1, according to the above lemma with a = b|α|+2, we then have
I1 � δ1/2∑
β≤|α|
∫ t
−r0‖|L(logμ)|Ri
β+1Lψ‖L2(�τ )
� δ1/2 Ib|α|+2(t)√E≤|α|+2(t, u) (8.21)
� δ1/2μ−b|α|+2m (t)
√E≤|α|+2(t, u).
Finally, the estimates (8.21), (8.7), (8.8) on I1, I2 and I3 together yield
‖Fα‖L2(�t )� ‖Fα‖L2(�−r0 )
+∫ t
−r0δ1/2μ
−b|α|+2m (τ )
(√E≤|α|+2(τ, u)
+μ−1/2m (τ )
√E≤|α|+2(τ, u)
)dτ + δ− 1
2μ−b|α|+2m (t)
×√∫ u
0F≤|α|+2(t, u
′)du′ + δ1/2μ−b|α|+2m (t)
√E≤|α|+2(t, u). (8.22)
123
On the formation of shocks for quasilinear wave equations
This implies
‖μ/d(Rαi trχ)‖L2(�t )
� δ1/2μ−b|α|+2m (t)
√E≤|α|+2(t, u)
+∫ t
−r0δ1/2μ
−b|α|+2−1/2m (τ )
√E≤|α|+2(τ, u)dτ
+ δ−1/2μ−b|α|+2m (t)
√∫ u
0F≤|α|+2(t, u
′)du′. (8.23)
8.2 Estimates on μ
The top order estimates on μ depend on the equation Lμ = m + μe. Since
m = −12d(c2)dρ Tρ and eμ = 1
2c2d(c2)dρ Lρ ·μ, it is visible that μ can be bounded
by the total energy on ψ , i.e. the Ek’s. However, to avoid loss of derivatives,we should not integrate Lμ directly.
On the formation of shocks for quasilinear wave equations
Then in view of (8.24), (8.26) and (8.27), we obtain the following propagationequation for F ′:
LF ′ + (trχ − 2μ−1Lμ)F ′ = −(1
2trχ − 2μ−1Lμ
)f ′ − 2μχ · /D
2μ + g′.
(8.29)
where
g′ =(
−/dμ + μ
c2dc2
dρ/dρ
)· (μ/dtrχ) + �
≤2≥−2 + O≤1
0 �≤2≥−2 + �
≤2≥0 .
(8.30)
Herewe have used the structure equation (2.34) to cancel the contribution fromthe term 1
2 Lρ(T trχ) + 12 Lρ(T trχ) in (8.25) and the term (Lμ) /�μ when we
write L(μ /�μ) = μL( /�μ) + L(μ) /�μ. We also remark that the L2 norm ofall derivatives on /divχ has been estimated from previous subsection. In such
a sense, it can also be considered as a �≤22 term and we use (2.33) to replace
/divχ by /dtrχ + · · · . The term�≤2≥−2 comes from the contribution of Lψ0Lψ0
andO≤10 �
≤2≥−2 comes from−1
4dc2dρ T Lρtrχ in (8.26). Since we already applied
T to Lψ0Lψ0 once in (8.26), instead of using flux F(t, u) as we did in the lastsubsection, we only need to use the energy E(t, u) to control the contributionof this term.
We set F ′α,l = μRi
α′T l /�μ − Ri
α′T l f ′ and |α′| + l = |α|. According to
(8.29), we have
LF ′α,l + (trχ − 2μ−1Lμ)F ′
α,l = −(1
2trχ − 2μ−1Lμ
)Ri
α′T l f ′
− 2μχ · /Lα′Ri/LlT /D2
μ + g′α′,l . (8.31)
where g′α′,l is given by
g′α′,l =
(−/dμ + μ
c2dc2
dρ/dρ
)· μ/d
(Ri
α′T l trχ
)+ l · �F ′
α,l−1 + (Ri )ZF ′α−1,l
+O≤|α′|+1−k O≤|α|−|α′|+1
≥2−2l+k + �≤|α|+2≥−2l−2 + O≤|α′|+1
−k �≤|α|+2−|α′|≥−2l−2+k + �
≤|α|+2≥−2l .
We remark that the second term of g′α′,l vanishes when l = 0 and� = [L, T ].
According to (8.31), we have
123
S. Miao, P. Yu
‖F ′α,l‖L2(�t )
� ‖F ′α,l‖L2(�−r0 )
+∫ t
−r0‖μ−1Lμ‖L∞(�τ )‖Ri
α′T l f ′‖L2(�τ )
dτ
+∫ t
−r0‖μχ · /LRi
α′/LlT /D2
μ‖L2(�τ )dτ +
∫ t
−r0‖g′
α′,l‖L2(�τ )dτ
= ‖F ′α‖L2(�−r0 )
+ I1 + I2 + I3.
We remark that we must multiply both sides δl to get the correct estimates.
We first deal with I1. Since f ′ = 12dc2dρ (−T Lρ + μ2
c2/�ρ), therefore, we have
Riα′T l f = −dc2
dρψ0Ri
α′T l+1Lψ0 + μ2
2c2dc2
dρψ0R
α′i T l /�ψ0 + �
≤|α|+2≥−2(l+2).
Compared to the first two terms, the last term on the right hand side above isof lower order with respect to the order of derivatives. Hence,
δl+1‖Riα′T l f ′‖L2(�τ )
� δ1/2√E≤|α|+2(τ, u) + δ3/2
√E≤|α|+2(τ, u)
(8.32)
This together with Lemma 8.1 yields
δl+1 I1 � μ−b|α|+2m (t)δ1/2
(√E≤|α|+2(t, u) + δ
√E≤|α|+2(t, u)
).
(8.33)
For I2, we can use elliptic estimates, i.e. to bound /D2μ by /�μ. This leads
to
δl+1 I2 � δl+2∫ t
−r0‖μRi
α′T l /�μ‖L2(�τ )
dτ
+∫ t
−r0δ3/2
√E≤|α|+2(τ, u) + δ3/2μ
−1/2m (τ )
√E≤|α|+2(τ, u)dτ
� δ
∫ t
−r0δl+1‖Fα,l‖L2(�τ )
+ δl+1‖Riα′T l f ‖L2(�τ )
+δ1/2(√
E≤|α|+2(τ, u) + μ−1/2m (τ )
√E≤|α|+2(τ, u)
)dτ.
(8.34)
We can skip the first two terms: The second term is already controlled by I1.While the first term will be eventually absorbed by Gronwall’s inequality. The
123
On the formation of shocks for quasilinear wave equations
last two terms come from the commutator between /� and T, Ri as well asusing Proposition 7.5.
For I3 := I ′3 + I ′′
3 , we first consider the contributions from the first line inthe expression of g′
α′,l , which are denoted by I ′3. We consider the cases l = 0
and l > 0 separately. For l = 0, we have
δ I ′3 � δ
∫ t
−r0
∥∥∥∥
(−/dμ + μ
c2dc2
dρ/dρ
)(μ/d(Ri
αtrχ))
+(| (Ri )Z | + |�|
)/dF ′|α|−1
∥∥∥∥L2(�τ )
dτ
� δ
∫ t
−r0‖μ/d
(Ri
αtrχ)
‖L2(�τ )+ δ‖/dF ′|α|−1‖L2(�τ )
dτ
+∫ t
−r0
(δ3/2
√E≤|α|+2(τ, u) + δ3/2μ
−1/2m (τ )
√E≤|α|+2(τ, u)
)dτ
The first two terms are bound by the top order estimates on trχ in the previoussubsection, therefore, for l = 0, we have
δ I ′3 �
∫ t
−r0δ3/2μ
−b|α|+2m (τ )
√E≤|α|+2(τ, u) + δ3/2μ
−b|α|+2−1/2m (τ )
×√E≤|α|+2(τ, u)dτ
+ δ1/2∫ t
−r0μ
−b|α|+2m (τ )
√∫ u
0F≤|α|+2(τ, u
′)du′dτ. (8.35)
For the l ≥ 1 case, we use (2.34), i.e. /LTχ = /∇⊗η + μ−1ζ ⊗η −c−1L(c−1μ)χ + c−1μθ⊗χ to rewrite T trχ . By taking the trace in (2.34),we obtain
μ/d(Rα′−1i T l trχ) = μRα′
i T l−1 /�μ + O≤|α|+1≥−2l+2,
where
δl+1‖O≤|α|+1≥−2l+2‖L2(�τ )
�δ3/2√E≤|α|+2(t ′, u)+δ3/2μ
−1/2m (τ )
√E≤|α|+2(τ, u).
We then conclude that (for l ≥ 1)
δl+1 I ′3 �
∫ t
−r0δ3/2μ
−b|α|+2m (τ )
√E≤|α|+2(τ, u) + δ3/2μ
−b|α|+2− 12
m (τ )
×√E≤|α|+2(τ, u)dτ. (8.36)
123
S. Miao, P. Yu
Now we discuss the contributions from the last three terms in the expressionof g′
α′,l , which are denoted by I ′′3 . In view of Propositions 7.4 and 7.5, the
termsO≤|α′|+1−k O≤|α|−|α′|+1
2−k can be absorbed byO≤|α′|+1−k �
≤|α|+2−|α′|≥−2l−2+k . We can
bound the last three terms of g′α′,l as follows
δl+1‖�≤|α|+2≥−2l−2‖L2(�t )
� δ1/2√E≤|α|+2(t, u),
δl+1‖O≤|α′|+1−k �
≤|α|+2−|α′|≥−2l−2+k ‖L2(�
ut )
� δ1/2√E≤|α|+2(t, u)
+δ1/2∫ t
−r0μ
−1/2m (t ′)
√E≤|α|+2(t
′, u)dt ′,
δl+1‖�≤|α|+2≥−2l ‖L2(�t )
� δ3/2√E≤|α|+2(t, u)
+δ3/2μ−1/2m (t)
√E≤|α|+2(t, u). (8.37)
Therefore we have the following estimates for I ′′3 :
δl+1 I ′′3 � δ1/2
∫ t
−r0μ
−b|α|+2m (t ′)
√E≤|α|+2(t ′, u)dt ′
+ δ1/2∫ t
−r0μ
−b|α|+2−1/2m (t ′)
√E≤|α|+2(t
′, u)dt ′. (8.38)
By combining the estimate (8.32) for f ′ and the estimates for (8.33), (8.34),(8.35), (8.36) and (8.38) for I1, I2, I3, we obtain
δl+1‖Rα′i T l /�μ‖L2(�t )
� δl+1‖Fα,l‖L2(�−r0 )
+ δ1/2μ−b|α|+2m (t)
(√E≤|α|+2(t, u) + μ
−b|α|+2m (t)
√E≤|α|+2(t, u)
)
+∫ t
−r0δ1/2μ
−b|α|+2m (τ )
√E≤|α|+2(τ, u)
+ δ1/2μ−b|α|+2−1/2m (τ )
√E≤|α|+2(τ, u)dτ
+ δ1/2∫ t
−r0μ
−b|α|+2m (τ )
√∫ u
0F≤|α|+2(τ, u
′)du′dτ. (8.39)
9 Commutator estimates
In this section, we shall estimate the error spacetime integrals for the contri-butions of commutators.
123
On the formation of shocks for quasilinear wave equations
Let ψ be a solution of the inhomogeneous wave equation �gψ = ρ and Zbe a vector field, one can commute Z with the equation to derive
�g (Zψ) = Zρ + 1
2trg
(Z)π · ρ + c2divg(Z) J (9.1)
where the vector field (Z) J is defined by
(Z) Jμ =((Z)πμν − 1
2gμν trg
(Z)π
)∂νψ.
We remark that the raising indices for (Z)πμν are with respect to the opticmetric g.
In applications,we use the above formulas for homogeneouswave equations�gψ = 0 and commute some commutation vector fields Zi ’s several times.Therefore, we need the following recursion formulas:
�gψn = ρn. ψn = Zψn−1, ρ1 = 0,
ρn = Zρn−1 + 1
2trg
(Z)π · ρn−1 + c2divg(Z) Jn−1,
(Z) Jμn−1 =((Z)πμν − 1
2gμν trg
(Z)π
)∂νψn−1.
(9.2)
Remark 9.1 When we derive energy estimates for �gψn = ρn , due to thevolume form of the conformal optic metric g, the integrands ρn appearing inthe error terms is slightly different from ρn . The rescaled source terms ρn aredefined as follows:
ρn = 1
c2μρn = Z ρn−1 + (Z)δ · ρn−1 + (Z)σn−1,
ρ1 = 0, (Z)σn−1 = μ · divg (Z) Jn−1,
(Z)δ = 1
2trg
(Z)π − μ−1Zμ + 2Z (log(c)) .
(9.3)
In view of (3.29) as well as the formula
tr(Z)π = ctr(Z)π = c(−μ−1(Z)πLL + tr(Z) /π
)
we have:∣∣∣(T )δ
∣∣∣ � 1,
∣∣∣(Q)δ
∣∣∣ � 1,
∣∣∣(R j )δ
∣∣∣ � δ. (9.4)
123
S. Miao, P. Yu
Then error spacetime integral corresponding to K0 = L and K1 = Lcontaining ρn are as follows:
−∫
Wtu
1
c2ρnLψndμg = −
∫
Wtu
ρnLψndtdudμ/g
−∫
Wtu
1
c2ρnLψndμg = −
∫
Wtu
ρnLψndtdudμ/g
We first consider the contribution of (Z)σn−1 in ρn . We write (Z)σn−1 in nullframe (L, L , ∂
∂θ A):
(Z)σn−1 = −1
2L((Z) Jn−1,L) − 1
2L((Z) Jn−1,L) + /div(μ(Z) /Jn−1)
−1
2L(c−2μ)(Z) Jn−1,L − 1
2trχ(Z) Jn−1,L − 1
2trχ(Z) Jn−1,L ,
Then with the following expressions for the components of (Z) Jn−1 in the nullframe:
(Z) Jn−1,L = −1
2tr(Z)
/π(Lψn−1) + (Z)Z · /dψn−1
(Z) Jn−1,L = −1
2tr(Z)
/π(Lψn−1) + (Z)Z · /dψn−1 − 1
2μ(Z)πLL(Lψn−1)
μ(Z) /J A
n−1 = −1
2(Z)
Z A(Lψn−1) − 1
2(Z)
ZA(Lψn−1)
+ 1
2((Z)πLL − μtr
(Z)/π)/d A
ψn−1 + μ(Y )
/πAB /d
Bψn−1
Based on the above expressions, we decompose:
(Z)σn−1 = (Z)σ1,n−1 + (Z)σ2,n−1 + (Z)σ3,n−1
where (Z)σ1,n−1 contains the products of components of (Z)π with the 2ndderivatives of ψn−1, (Z)σ2,n−1 contains the products of the 1st derivatives of(Z)π with the 1st derivatives of ψn−1, and (Z)σ3,n−1 contains the other lowerorder terms. More specifically, we have:
123
On the formation of shocks for quasilinear wave equations
(Z)σ1,n−1 = 1
2tr(Z)
/π
(LLψn−1 + 1
2tr/gχLψn−1
)
+1
4(μ−1(Z)πLL)L
2ψn−1
−(Z)Z · /dLψn−1 − (Z)
Z · /dLψn−1
+1
2(Z)πLL /�ψn−1 + μ
(Z)/π · /D2
ψn−1 (9.5)
(Z)σ2,n−1 = 1
4L(tr
(Z)/π)Lψn−1 + 1
4L(tr
(Z)/π)Lψn−1
+ 1
4L(μ−1(Z)πLL)Lψn−1
− 1
2/LL
(Z)Z · /dψn−1 − 1
2/LL
(Z)Z · /dψn−1
− 1
2/div
(Z)Z Lψn−1 − 1
2/div
(Z)Z Lψn−1
+ 1
2/d(Z)πLL /dψn−1 + /div(μ
(Z)/π) · /dψn−1 (9.6)
and
(Z)σ3,n−1 =(1
4trχ tr
(Z)/π + 1
4trχ(μ−1(Z)πLL)
+1
2(Z)
Z · /d(c−2μ)
)Lψn−1 − 1
4(L log(c−1))tr
(Z)/πLψn−1
−((
1
2trχ + L(c−2μ)
)(Z)
Z + 1
2trχ
(Z)Z
)/dψn−1 (9.7)
With these expressions for (Z)σn−1, we are able to investigate the structureof ρn . Basically, we want to use the recursion formulas in (9.3) to obtain arelatively explicit expression for ρn .
On the other hand, for the energy estimates, we consider the followingpossible ψn:
ψn = Rα+1i ψ, ψn = Rα′
i T l+1ψ, ψn = QRα′i T lψ
Here ψn is the nth order variation and n = |α| + 1 = |α′| + l + 1. Whileψ is any first order variation. The reason that we can always first apply T ,then Ri , and finally a possible Q is that the commutators [Ri , T ], [Ri , Q] and[T, Q] are one order lower than RiT, T Ri ; QRi , Ri Q; QT, T Q respectively.
123
S. Miao, P. Yu
Moreover, the commutators [Ri , T ], [Ri , Q] and [T, Q] are tangent to St,u .Since we let Q be the last possible commutator, there will be no Q’s in ψn−1in the second term on the right hand side of (9.5). Therefore we only needto commute Q once. Now suppose that we consider the n = |α| + 2th ordervariations of the following form:
ψ|α|+2 := Z|α|+1 · · · Z1ψ
We have the inhomogeneous wave equation:
�gψ|α|+2 = ρ|α|+2
As we pointed out in Remark 9.1, we define:
ρ|α|+2 = μ
c2ρ|α|+2
Then by a induction argument, the corresponding inhomogeneous term ρ|α|+2is given by:
Consider an arbitrary term in this sum. There is a total of k derivatives withrespect to the commutators acting on (Z)σ|α|−1+k . In view of the fact that(Z)σ|α|−1+k has the structure described in (9.5)–(9.7), in considering the partialcontribution of each term in (Z)σ1,|α|−1+k , if the factor which is a component of(Z)π receives more than
[ |α|+12
]derivatives with respect to the commutators,
then the factor which is a 2nd order derivative ofψ|α|+1−k receives at most k−[ |α|+12
]− 1 order derivatives of commutators, thus corresponds to a derivative
of the ψ of order at most: k − [ |α|+12
]+ 1+|α|− k = [ |α|2
]+ 1, therefore thisfactor is bounded in L∞(�
ut ) by the bootstrap assumption.Also, in considering
the partial contribution of each term in (Z)σ2,|α|+1−k , if the factor which is a 1stderivative of (Z)π receivesmore than
[ |α|+12
]−1 derivatives with respect to thecommutators, then the factor which is a 1st derivative of ψ|α|+1−k receives atmost k−[ |α|+1
2
]derivatives with respect to the commutators, thus corresponds
to a derivative of theψα of order at most k− [ |α|+12
]+1+|α|−k = [ |α|2
]+1,therefore this factor is again bounded in L∞(�
ut ) by the bootstrap assumption.
123
On the formation of shocks for quasilinear wave equations
Similar considerations apply to (Z)σ3,|α|+1−k . We conclude that for all theterms in the sum in (9.8) of which one factor is a derivative of the (Z)π oforder more than
[ |α|+12
], the other factor is then a derivative of the ψα of order
at most[ |α|
2
]+1 and is thus bounded in L∞(�ut ) by the bootstrap assumption.
Of these terms we shall estimate the contribution of those containing the toporder spatial derivatives of the optical entities in the next subsection. Beforewe give the estimates for the contribution of the lower order optical terms tothe spacetime integrals:
− δ2k∫
Wtu
ρ≤|α|+2Lψ≤|α|+2dtdudμ/g,
− δ2k∫
Wtu
ρ≤|α|+2Lψ≤|α|+2dtdudμ/g,
(9.9)
we investigate the behavior of these integrals with respect to δ. Here k isthe number of T s in string of commutators. For the multiplier K1 = L , theassociated energy inequality is
E≤|α|+2(t, u) + F≤|α|+2(t, u) + K≤|α|+2(t, u)
� E≤|α|+2(−r0, u) +∫
Wtu
Q1,≤|α|+2. (9.10)
The quantities K≤|α|+2(t, u) are defined similar as K (t, u):
K≤|α|+2(t, u) :=∑
|α′|≤|α|+1
δl′K (t, u)[Zα′
ψ],
Again, l ′ is the number of T ’s in Zα′.
In Q1,≤|α|+2, there are contributions from the deformation tensors of twomultipliers, which has been treated in Sect. 6. There are also contributionsfrom the deformation tensors of commutators, which are given by (9.8). Nowwe investigate the terms which are not top order optical terms, namely, theterms containing χ and μ of order less than |α| + 2. In view of the discussionin Sect. 6, the left hand side of (9.10) is of order δ, so we expect these lowerorder terms in the second integral of (9.9) is of order δ. In fact, the integrationon Wt
u gives us a δ and the multiplier δk Lψ≤|α|+2 is of order δ1/2. To see the
behavior of δkσ , we look at σ1 as an example. Let k′ be the number of T sapplied to (LLψl + 1
2 trχLψl). (4.1) implies
123
S. Miao, P. Yu
δk′(LLψl + 1
2trχLψl
)∼ δ1/2 + δk
′ρl .
Since ρ1 = 0, an induction argument implies that
δk′(LLψl + 1
2trχLψl
)∼ δ1/2.
Then in view of (3.24), (3.26) and (3.29), the first term in σ1 behaves like δ1/2.Following the same procedure, one sees straightforwardly that all the otherterms in σ1, σ2 and σ3 behave like δ1/2 (one keeps in mind that if Z = T ,then we multiplier a δ with the corresponding deformation tensor.) except theterm L
(tr(Z) /π
)Lψl . For this term we use the argument deriving (3.28) and
Proposition 7.1 to see actually we have:
‖L(tr(Q) /π)‖L∞(�ut )
� δ.
This completes the discussions for σ associated to K1.The same argument applies to the energy inequality associated to K0:
and we conclude that the lower order optical terms in the error spacetimeintegrals have one more power in δ than the energies on the left hand side.
Now we summarize the spacetime error estimates for the terms whichcome from the L2 norms of the lower order optical quantities. In the proof
of Propositions 7.4 and 7.5, we use μ−1/2m (t)
√E≤|α|+2(t, u) to control the L
2
norm∑
α′≤|α|+1 ‖/dZα′ψ‖L2(�
ut ). Now we just keep this L2 norm as it is. This
together with the bootstrap assumptions on the L∞ norms of the variationsimplies that the contributions from the L2 norms of lower order optical termsare bounded as:
∫ t
−r0
⎛
⎝∑
|α′|≤|α|+1
δ1/2+l ′‖/dZα′i ψ‖L2(�
ut ′ )
+ δ
√E≤|α|+2(t ′, u)
⎞
⎠ dt ′
· δ1/2∫ t
−r0‖Lψ|α|+2‖L2(�
ut ′ )dt ′
123
On the formation of shocks for quasilinear wave equations
�∫ t
−r0
⎛
⎝∑
|α′|≤|α|+1
δ2+2l ′‖/dZα′i ψ‖2
L2(�ut ′ )
+ δ2E≤|α|+2(t′, u)
⎞
⎠ dt ′
+∫ u
0F≤|α|+2(t, u
′)du′ for K1
and
∫ t
−r0
⎛
⎝∑
|α′|≤|α|+1
δ1/2+l ′‖/dZα′i ψ‖L2(�
ut ′ )
+ δ
√E≤|α|+2(t ′, u)
⎞
⎠ dt ′
· δ1/2∫ t
−r0‖Lψ|α|+2‖L2(�
ut ′ )dt ′
�∫ t
−r0
⎛
⎝∑
|α′|≤|α|+1
δ1+2l ′‖/dZα′i ψ‖2
L2(�ut ′ )
+ δE≤|α|+2(t′, u)
⎞
⎠ dt ′
+ δ
∫ t
−r0E≤|α|+2(t
′, u)dt ′ for K0
where l ′ is the number of T ’s in the string of Zα′i . Therefore we obtain the
following error estimates for the lower order optical terms:
∫ t
−r0δ2E≤|α|+2(t
′, u)dt ′+∫ u
0F≤|α|+2(t, u
′)du′+δ2K≤|α|+2(t, u) for K1
(9.12)
and
∫ t
−r0δE≤|α|+2(t
′, u)dt ′ + δK≤|α|+2(t, u) for K0 (9.13)
Next we consider the case in which the deformation tensors receive less deriv-atives with respect to the commutators than the variations in the expression for(Z)σl . More specifically, we consider the terms in the sum (9.8) in which there
are at most [ |α|+12 ] derivatives hitting the deformation tensor
(Z)/π , thus the
spatial derivatives on χ is at most [ |α|+12 ] and the spatial derivatives on μ is at
most [ |α|+12 ]+ 1, which are bounded in L∞(�
ut ) by virtue of Propositions 7.1
and 7.2. Using the inequality ab ≤ εa2+ 1εb2,we have the following estimates
for these contributions:
123
S. Miao, P. Yu
∫ t
−r0δ2E≤|α|+2(t
′, u)dt ′ + δ−1/2∫ u
0F≤|α|+2(t, u
′)du′
+ δ1/2K≤|α|+2(t, u) +∫ t
−r0E≤|α|+2(t
′, u)dt ′ for K1 (9.14)
∫ t
−r0δ1/2E≤|α|+2(t
′, u)dt ′ + δ−1/2∫ u
0F≤|α|+2(t, u
′)du′
+ δ1/2K≤|α|+2(t, u) for K0 (9.15)
Here we estimate the terms involving L2ψn−1 and Lψn−1 in terms of flux.
9.2 Top order optical estimates
Now we estimate the contributions from the top order optical terms to theerror spacetime integrals. In estimating the top order optical terms, we need tochoose the power of μm(t) large enough. Therefore from this subsection on,we will use C to denote an absolute positive constant so that one can see thelargeness of the power of μm(t) more clearly.
The top order optical terms come from the term inwhich all the commutatorshit the deformation tensors in the expression of (Z1)σ1, namely, the term:
(Z|α|+1 + (Z|α|+1)δ) · · · (Z2 + (Z2)δ)(Z1)σ1
more precisely, in:
Z|α|+1 · · · Z2
(−1
2L((Z1) J1,L) − 1
2L((Z1) J1,L) + /div(μ(Z1) /J )
)
when the operators L , L, /div hit the deformation tensors in the expression of(Z1) J .
Now we consider the top order variations:
Rα+1i ψ, Rα′
i T l+1ψ, QRα′i T lψ
where |α| = Ntop − 1 and |α′| + l + 1 = |α| + 1. Then the correspondingprincipal optical terms are:
ρ|α|+2(Rα+1i ψ) := 1
c(Rα+1
i trχ ′) · Tψ,
ρ|α|+2(Rα′i T l+1ψ) := 1
c(Rα′
i T l /�μ) · Tψ
ρ|α|+2(QRα′i T lψ) := tμ
c
(/dRα′
i trχ ′) · /dψ + tμ
c(/dRα′
i /�μ) · Lψ, if l = 0
123
On the formation of shocks for quasilinear wave equations
ρ|α|+2(QRα′i T lψ) := tμ
c
(/dRα′
i T l−1 /�μ) · /dψ + tμ
c(/dRα′
i T l /�μ) · Lψ,if l ≥ 1
Here we used the structure equation (2.34)
/LT trχ = /�μ + O≤1≥0
Now we briefly investigate the behavior of the above terms with respect to δ.Note that Rα
i Qψ has the same behavior as Rα+1i ψ with respect to δ, while for
their corresponding toporder optical terms tμc
(/dRα
i trχ′)·/dψ and 1
c
(Rα+1i trχ ′)·
Tψ , the former behaves better than the latter with respect to δ:
∣∣∣∣tμ
c(/dRα
i trχ′) · /dψ
∣∣∣∣ ∼ |μ(Rα+1
i trχ ′)|δ1/2,∣∣∣∣1
c(Rα+1
i trχ ′) · Tψ∣∣∣∣ ∼ |(Rα+1
i trχ ′)|δ−1/2
We see that not only the former behaves better with respect to δ, but alsohas an extra μ, which makes the behavior even better when μ is small. Thismeans that we only need to estimate the contribution of 1
c
(Rα+1i trχ ′) · Tψ .
The same analysis applies to the terms involving Lψ as well as the compar-ison between Rα′
i Tψ and QRα′i Tψ , which correspond to 1
c
(Rαi/�μ) · Tψ
and tμc (R
α′+1i
/�μ) · /dψ . So in the following, we do not need to estimate thecontributions corresponding to the variations containing a Q.
9.2.1 Contribution of K0
In this subsection we first estimate the spacetime integral:
∫
Wtu
1
c|Rα+1
i trχ ′||Tψ ||LRα+1i ψ |dt ′du′dμ/g
�∫ t
−r0sup�
ut ′(μ−1|Tψ |)‖μ/dRα
i trχ′‖L2(�
ut ′ )
‖LRα+1i ψ‖L2(�
ut ′ )dt ′ (9.16)
By (8.23), we have:
‖μ/dRαi trχ
′‖L2(�ut )
� δ1/2μ−b|α|+2m (t)
√E≤|α|+2(t, u)
123
S. Miao, P. Yu
+∫ t
−r0δ1/2μ
−b|α|+2−1/2m (τ )
√E≤|α|+2(τ, u)dτ
+ δ−1/2μ−b|α|+2m (t)
√∫ u
0F≤|α|+2(t, u
′)du′.
By the monotonicity of E≤|α|+2(t, u) in t , we have
‖μ/dRαi trχ
′‖L2(�ut )
� δ1/2μ−b|α|+2m (t)
√E≤|α|+2(t, u)
+ δ1/2√E≤|α|+2(t, u)
∫ t
−r0μ
−b|α|+2−1/2m (t ′)dt ′
+ δ−1/2
√∫ u
0F≤|α|+2(t, u
′)du′μ−b|α|+2m (t) (9.17)
Without loss of generality, here we assume that there is a t0 ∈ [−r0, t∗) suchthatμm(t0) = 1
10 andμm(t ′) ≥ 110 for t ≤ t0. If there is no such t0 in [−r0, t∗),
thenμm(t) has an absolute positive lower bound for all [−r0, t∗) and it is clearto see that the following argument simplifies and also works in this case. Inview of part (1′) and part (2) in Lemma 8.1 and the fact μm(t ′) ≥ 1
10 fort ′ ∈ [−r0, t0], we have
∫ t0
−r0μ
−b|α|+2−1/2m (t ′)dt ′ � μ
−b|α|+2+1/2m (t0) ≤ μ
−b|α|+2+1/2m (t),
∫ t
t0μ
−b|α|+2−1/2m (t ′)dt ′ � 1
(b|α|+2 − 1/2
)μ−b|α|+2+1/2m (t).
Therefore the second term in (9.17) are bounded by:
δ1/2μ−b|α|+2+1/2m (t)
√E≤|α|+2(t, u)
Substituting this in (9.16), and using the fact that |Tψ | � δ−1/2, we seethat the spacetime integral (9.16) is bounded by:
∫ t
−r0μ
−b|α|+2−1m (t ′)
√E≤|α|+2(t ′, u)‖LRα+1
i ψ‖L2(�ut ′ )dt ′
+∫ t
−r0μ
−b|α|+2−1/2m (t ′)
√E≤|α|+2(t
′, u)‖LRα+1i ψ‖L2(�
ut ′ )dt ′
+∫ t
−r0δ−1μ
−b|α|+2−1m (t ′)
√∫ u
0F≤|α|+2(t
′, u′)du′‖LRα+1i ψ‖L2(�
ut ′ )dt ′
123
On the formation of shocks for quasilinear wave equations
For the factor ‖LRα+1i ψ‖L2(�
ut ), we bound it as:
‖LRα+1i ψ‖L2(�
ut )
≤ √E|α|+2(t, u) ≤ μ−b|α|+2m (t)
√E≤|α|+2(t, u)
Then the spacetime integral (9.16) is bounded by:
∫ t
−r0μ
−2b|α|+2−1m (t ′)E≤|α|+2(t
′, u)dt ′
+∫ t
−r0μ
−2b|α|+2−1/2m (t ′)
√E≤|α|+2(t
′, u)√E≤|α|+2(t ′, u)dt ′
+ δ−1∫ t
−r0μ
−2b|α|+2−1m (t ′)
√∫ u
0F≤|α|+2(t
′, u′)du′√E≤|α|+2(t ′, u)dt ′
Splitting the above integrals as∫ t0−r0
+ ∫ tt0 , we see that the “non-shock” part∫ t0−r0
is bounded by:
∫ t0
−r0μ
−2b|α|+2m (t ′)
(E≤|α|+2(t
′, u) + E≤|α|+2(t′, u)
+δ−2∫ u
0F≤|α|+2(t
′, u′)du′)dt ′ (9.18)
using part (2) of Lemma 8.1.The “shock” part
∫ tt0is bounded by:
C
2b|α|+2μ
−2b|α|+2m (t)E≤|α|+2(t, u)
+ C(2b|α|+2 − 1/2
)μ−2b|α|+2+1/2m (t)E≤|α|+2(t, u)
+ δ−2 C(2b|α|+2
)μ−2b|α|+2m (t)
∫ u
0F≤|α|+2(t, u
′)du′ (9.19)
using part (1′) of Lemma 8.1.
Remark 9.2 The boxed term is from the estimates for the top order term Fα .In view of (8.3), the number of top order terms contributed by the variationsis independent of δ and |α|, so is the constant C in the boxed term. Later onin the top order energy estimates we will choose b|α|+2 in such a way thatCbtop
≤ 110 . (The purpose of doing this is to make sure that this term can be
123
S. Miao, P. Yu
absorbed by the left hand side the energy inequality.) Therefore we can choosebtop = {10, C
20 }. In particular btop is independent of δ.Next we consider the spacetime integral:
δ2l+2∫
Wtu
1
c|Rα′
i T l /�μ||Tψ ||LRα′T l+1ψ |dt ′du′dμ/g
� δ2l+2∫ t
−r0sup�
ut ′
(μ−1|Tψ |)‖μRα′
i T l /�μ‖L2(�ut ′ )
‖LRα′T l+1ψ‖L2(�
ut ′ )dt ′
(9.20)
By (8.39) and the monotonicity of E≤|α|+2(t, u), E≤|α|+2(t, u) and F≤|α|+2(t, u) in t , we have:
δl+1‖μRα′i T l /�μ‖L2(�
ut ′ )
≤ Cδl+1‖Fα,l‖L2(�u−r0
)
+ δ1/2μ−b|α|+2m (t)
√E≤|α|+2(t, u) + δ1/2μ
−b|α|+2m (t)
√E≤|α|+2(t, u)
+∫ t
−r0
(δ1/2μ
−b|α|+2m (t ′)
√E≤|α|+2(t ′, u)
+ δ1/2μ−b|α|+2−1/2m (t ′)
√E≤|α|+2(t
′, u))dt ′
+ δ1/2∫ t
−r0μ
−b|α|+2m (t ′)
√∫ u
0F≤|α|+2(t
′, u′)du′dt ′ (9.21)
As before, we have:
∫ t
−r0μ
−b|α|+2m (t ′)dt ′ ≤ Cμ
−b|α|+2+1m (t),
∫ t
−r0μ
−b|α|+2−1/2m (t ′)dt ′ ≤ Cμ
−b|α|+2+1/2m (t)
Therefore
δl+1‖μRα′i T l /�μ‖L2(�
ut ′ )
≤ Cδl+1‖Fα,l‖L2(�u−r0
)
+Cδ1/2μ−b|α|+2m (t)
√E≤|α|+2(t, u) + Cδ1/2μ
−b|α|+2m (t)
√E≤|α|+2(t, u)
+Cδ1/2μ−b|α|+2+1m (t)
√∫ u
0F≤|α|+2(t, u
′)du′ (9.22)
123
On the formation of shocks for quasilinear wave equations
Again, we estimate the other factor ‖LRα′i T l+1ψ‖L2(�
ut )by:
δl+1‖LRα′i T l+1ψ‖L2(�
ut )
≤ √E≤|α|+2(t, u) ≤ μ−b|α|+2m (t)
√E≤|α|+2(t, u)
Here we also split the spacetime integral (9.20) into “shock part∫ tt0” and
“non-shock part∫ t0−r0
”. We first estimate the “shock part”. Due to the estimate
|Tψ | � δ−1/2, the contribution of the first term in (9.21) to the spacetimeintegral (9.20) is bounded by:
∫ t
t0μ
−b|α|+2−1m (t ′)
√E≤|α|+2(t, u)‖LRα′
i T l+1ψ‖L2(�ut ′ )dt ′
≤ C∫ t
t0μ
−2b|α|+2−1m (t ′)E≤|α|+2(t
′, u)dt ′
≤ C E≤|α|+2(t, u)∫ t
t0μ
−2b|α|+2−1m (t ′)dt ′
≤ C
2b|α|+2μ
−2b|α|+2m (t)E≤|α|+2(t, u)
Therefore, the contributions from the first two terms in (9.21) are bounded by:
C
2b|α|+2μ
−2b|α|+2m (t)
[E≤|α|+2(t, u) + E≤|α|+2(t, u)
]
and the contributions from the rest terms in (9.21) are bounded by:
C(2b|α|+2 − 1/2
)μ−2b|α|+2+1/2m (t)E≤|α|+2(t, u)
+ C(2b|α|+2 − 1
)μ−2b|α|+2+1m (t)
[E≤|α|+2(t, u) +
∫ u
0F≤|α|+2(t, u
′)du′]
Therefore “shock-part” of the spacetime integral (9.20) is bounded by:
C
2b|α|+2μ
−2b|α|+2m (t)
[E≤|α|+2(t, u) + E≤|α|+2(t, u)
]
+ C(2b|α|+2 − 1
)μ−2b|α|+2+1m (t)
∫ u
0F≤|α|+2(t, u
′)du′ (9.23)
As before, for the “non-shock part”, which mean t ′ ∈ [−r0, t0], we have apositive lower bound for μm(t ′), then the spacetime integral (9.20) is boundedby:
123
S. Miao, P. Yu
∫ t0
−r0μ
−2b|α|+2m (t ′)
[E≤|α|+2(t, u) + E≤|α|+2(t
′, u)
+∫ u
0F≤|α|+2(t
′, u′)du′]dt ′ (9.24)
which will be treated by Gronwall.Finally, using the inequality ab ≤ 1
2b2 + 1
2b2, the initial contribution
‖Fα,l‖L2(�−r0 )is bounded by
δ2l+2‖Fα,l‖2L2(�−r0 )+∫ t
−r0μ
−2b|α|+2−1m (t ′)E≤|α|+2(t
′, u)dt ′. (9.25)
The second term above has already been estimated.
9.2.2 Contribution of K1
In this subsection we estimate the contributions of top order optical termsassociated to K1. We start with the following absolute value of a spacetimeintegral:
∣∣∣∣∣
∫
Wtu
1
c(Rα+1
i trχ ′) · (Tψ) · (LRα+1i ψ)dt ′du′dμ/g
∣∣∣∣∣
(9.26)
Since
/g = 1
c/g, dμ/g = 1
cdμ/g
the above spacetime integral can be written as:
∫
Wtu
(Rα+1i trχ ′) · (Tψ) · (LRα+1
i ψ)dt ′du′dμ/g (9.27)
Let us set:
F(t, u) =∫
St,uf dμ/g
then
∂F
∂t=∫
St,u
(L f + trχ f
)dμ/g
123
On the formation of shocks for quasilinear wave equations
So we may convert the spacetime integral into two hypersurface integrals:∫
Wtu
(L f + trχ f
)dμ/gdu
′dt ′ =∫
�ut
f dμ/gdu′ −∫
�u−r0
f dμ/gdu′
So we write the spacetime integral (9.27) as:∫
Wtu
(Rα+1i trχ ′) · (Tψ) · (LRα+1
i ψ + trχRα+1i ψ)dt ′du′dμ/g
−∫
Wtu
trχ(Rα+1i trχ ′) · (Tψ) · (Rα+1
i ψ)dt ′du′dμ/g
=∫
Wtu
(L + trχ
)[(Rα+1
i trχ ′) · (Tψ) · (Rα+1i ψ)
]dt ′du′dμ/g
−∫
Wtu
(L + trχ)[(Rα+1
i trχ ′) · (Tψ)]
· (Rα+1i ψ)dt ′du′dμ/g
We conclude that (9.27) equals:∫
�ut
(Rα+1i trχ ′) · (Tψ) · (Rα+1
i ψ)du′dμ/g
−∫
�u−r0
(Rα+1i trχ ′) · (Tψ) · (Rα+1
i ψ)du′dμ/g
−∫
Wtu
(L + trχ)[(Rα+1
i trχ ′) · (Tψ)]
· (Rα+1i ψ)dt ′du′dμ/g
We first consider the hypersurface integral which, integrating by parts, equals:
−H0 − H1 − H2
where:
H0 =∫
�ut
(Rαi trχ) · (Tψ) · (Rα+2
i ψ)du′dμ/g
H1 =∫
�ut
(Rαi trχ
′) · (RiTψ) · (Rα+1i ψ)du′dμ/g
H2 =∫
�ut
(Tψ) · (Rα+1i ψ) · (Rαtrχ ′)
(1
2tr(Ri )
/π
)du′dμ/g
Since we shall bound both Tψ and RiTψ in L∞ norm, compared to H0, H1is a lower order term with respect to the order of derivatives. While for H2, weuse the estimate:
123
S. Miao, P. Yu
|tr(Ri ) /π | ≤ Cδ
to see that it is a lower order term with respect to both the behavior of δ andthe order of derivatives compared to H0. This analysis tells us that we onlyneed to estimate H0.
|H0| ≤∫
�ut
|Tψ ||Rαi trχ
′||/dRα+1i ψ |du′dμ/g
≤ Cδ−1/2‖Rαi trχ
′‖L2(�ut )
‖/dRα+1i ψ‖L2(�
ut )
≤ C∫ t
−r0
(μ
−1/2m (t ′)
√E |α|+2(t
′, u))dt ′ · μ−1/2
m (t)√E |α|+2(t, u)
by Proposition 7.4. Then in terms of modified energies, we have:
|H0| ≤ C∫ t
−r0μ
−b|α|+2−1/2m (t ′)
√E≤|α|+2(t
′, u)dt ′
·μ−b|α|+2−1/2m (t)
√E≤|α|+2(t, u)
Again, we consider the “shock part∫ t0−r0
” and “non-shock part∫ tt0”, which
are denoted by HS0 and HN
0 , separately.In the “shock part”, by the monotonicity of E≤|α|+2(t), we have:
|HS0 | ≤ C
√E≤|α|+2(t, u)
∫ t
t0μ
−b|α|+2−1/2m (t ′)dt ′ · μ−b|α|+2−1/2
m (t)
√E≤|α|+2(t, u)
≤ C E≤|α|+2(t, u)μ−b|α|+2−1/2m (t)
∫ t
t0μ
−b|α|+2−1/2m (t ′)dt ′
≤ C(b|α|+2 − 1/2
)μ−b|α|+2−1/2m (t)E≤|α|+2(t, u) · μ−b|α|+2+1/2
m (t)
= C(b|α|+2 − 1/2
)μ−2b|α|+2m (t)E≤|α|+2(t, u)
The estimates for the “non-shock part” is more delicate.
|HN0 | ≤ C
∫ t0
−r0μ
−b|α|+2−1/2m (t ′)
√E≤|α|+2(t
′, u)dt ′
·μ−b|α|+2−1/2m (t)
√E≤|α|+2(t, u)
123
On the formation of shocks for quasilinear wave equations
Now for t ′ ∈ [−r0, t0], μ−1m (t ′) ≤ 10. This means we have the following
estimate for the above integral:
∫ t0
−r0μ
−b|α|+2−1/2m (t ′)
√E≤|α|+2(t
′, u)dt ′
=∫ t0
−r0μ−1m (t ′)μ−b|α|+2+1/2
m (t ′)√E≤|α|+2(t
′, u)dt ′
≤ C∫ t0
−r0μ
−b|α|+2+1/2m (t ′)
√E≤|α|+2(t
′, u)dt ′
≤ C∫ t0
−r0
√E≤|α|+2(t
′, u)dt ′ · μ−b|α|+2+1/2m (t)
where in the last step, we have used part (2) of Lemma 8.1.Therefore
|HN0 | ≤ C
∫ t0
−r0
√E≤|α|+2(t
′, u)dt ′ · μ−2b|α|+2m (t)
√E≤|α|+2(t, u)
≤ εμ−2b|α|+2m (t)E≤|α|+2(t, u) + Cεμ
−2b|α|+2m (t)
∫ t0
−r0E≤|α|+2(t
′, u)dt ′
Here ε is a small absolute constant to be determined later.We obtain the following estimate for |H0|:
|H0| ≤ C(b|α|+2 − 1/2
)μ−2b|α|+2m (t)E≤|α|+2(t, u)
+Cεμ−2b|α|+2m (t)
∫ t0
−r0E≤|α|+2(t
′, u)dt ′ + εμ−2b|α|+2m (t)E≤|α|+2(t, u)
(9.28)
Next we consider the spacetime integral:
∫
Wtu
(L + trχ)[(Rα+1
i trχ) · (Tψ)]
· (Rα+1i ψ)dt ′du′dμ/g
=∫
Wtu
((L + trχ)(Rα+1
i trχ ′)) · (Tψ) · (Rα+1
i ψ)dt ′du′dμ/g
+∫
Wtu
(Rα+1i trχ ′) · L(Tψ) · (Rα+1
i ψ)dt ′du′dμ/g := I + I I
123
S. Miao, P. Yu
For the second term in the above sum, by the fact that:
|L(Tψ)| ≤ Cδ−1/2
we have:
|I I | ≤ Cδ−1/2∫ t
−r0‖/dRα
i trχ′‖L2(�
ut ′ )
‖Rα+1i ψ‖L2(�
ut ′ )dt ′
On the other hand, by Lemma 7.3, we have:
‖Rα+1i ψ‖L2(�
ut ′ )
≤ Cδ(E|α|+2(t
′, u) + E |α|+2(t′, u)
)
which gives:
|I I | ≤ Cδ1/2∫ t
−r0‖/dRα
i trχ′‖L2(�
ut ′ )(E|α|+2(t
′, u) + E |α|+2(t′, u)
)dt ′
This has a similar form as (9.16), and moreover, it has an extra δ, which isconsistent with the order of E |α|+2(t, u). (We already took into account theeffect of L(Tψ).) So this term is already handled.
To estimate |I |, we first rewrite:
(L + trχ)(Rα+1i trχ ′) = Ri (L + trχ)(Rα
i trχ′) + (Ri )Z Rα
i trχ′
− Ri (trχ′)Rα
i trχ′ + l.o.t.
The contribution of the second term is:∫
Wtu
|Tψ ||(Ri )Z ||/dRαi trχ
′||Rα+1i ψ |dt ′du′dμ/g
By the estimates:
|(Ri )Z | ≤ Cδ, |Tψ | ≤ Cδ−1/2
this contribution is bounded by:
δ1/2∫ t
−r0‖/dRα
i trχ′‖L2(�
ut ′ )
‖Rα+1i ψ‖L2(�
ut ′ )dt ′
which is similar to the estimate for |I I | and has an extra δ, so this is a lowerorder term.
123
On the formation of shocks for quasilinear wave equations
By the estimate
|Ri trχ | ≤ Cδ,
the contribution of the third term is bounded by:
Cδ∫
Wtu
|Rαi trχ
′||Tψ ||Rα+1i ψ |dt ′du′dμ/g
≤ Cδ1/2∫ t
−r0‖Rα
i trχ′‖L2(�
ut ′ )
‖Rα+1i ψ‖L2(�
ut ′ )dt ′
By Proposition 7.4 and Lemma 7.3, this is bounded in terms of modifiedenergies by:
δ2∫ t
−r0μ
−b|α|+2−1/2m (t ′)
√E≤|α|+2(t
′, u) · μ−b|α|+2m (t ′)
×(√
E≤|α|+2(t ′, u) +√E≤|α|+2(t
′, u))dt ′
≤ Cδ2(E≤|α|+2(t, u) + E≤|α|+2(t, u)
) ∫ t
−r0μ
−2a−1/2m (t ′)dt ′
≤ Cδ2μ−2b|α|+2+1/2m (t)
(E≤|α|+2(t, u) + E≤|α|+2(t, u)
)(9.29)
Again, we used Lemma 8.1 in the last step.Now we are left with the spacetime integral:
∫
Wtu
(Ri (L + trχ)(Rα
i trχ′))
· (Tψ) · (Rα+1i ψ)dt ′du′dμ/g
Integrating by parts, this equals:
−∫
Wtu
((L + trχ)(Rαi trχ
′)) · (Tψ) · (Rα+2i ψ)dt ′du′dμ/g
−∫
Wtu
((L + trχ)(Rαi trχ
′)) ·(RiTψ + 1
2tr(Ri )
/π
)· (Rα+1
i ψ)dt ′du′dμ/g
:= −V1 − V2
By the estimate:
∣∣∣tr
(Ri )/π
∣∣∣ ≤ Cδ
123
S. Miao, P. Yu
we see that:∣∣∣∣RiTψ + 1
2tr(Ri )
/π
∣∣∣∣ ≤ Cδ−1/2, |Tψ | ≤ Cδ−1/2
So compared toV1,V2 is a lower order term, andweuseLemma7.3 to estimate:
‖Rα+1i ψ‖L2(�
ut )
≤ Cδ(√
E|α|+2(t, u) +√E |α|+2(t, u)
)
So we only need to estimate |V1|, which is bounded by:∫
Wtu
∣∣∣∣
(L + 2
t − u
)(Rα
i trχ′)∣∣∣∣ |Tψ ||Rα+2
i ψ |dt ′du′dμ/g
+∫
Wtu
|trχ ′||Rαi trχ
′||Tψ ||Rα+2i ψ |dt ′du′dμ/g := V11 + V12
By the estimates:
|trχ ′| ≤ Cδ, |Tψ | ≤ Cδ−1/2
and Proposition 7.4, V12 is bounded in terms of modified energies by:
δ1/2∫ t
−r0‖Rα
i trχ′‖L2(�
ut ′ )
‖/dRα+1i ψ‖L2(�
ut ′ )dt ′ ≤ Cδ
∫ t
−r0μ
−b|α|+2−1/2m (t ′)
√E≤|α|+2(t
′, u)μ−b|α|+2−1/2m (t ′)
√E≤|α|+2(t
′, u)dt ′
≤ Cδ E≤|α|+2(t, u)∫ t
−r0μ
−2b|α|+2−1m (t ′)dt ′≤Cδμ
−2b|α|+2m (t)E≤|α|+2(t, u).
(9.30)
In the last step we used Lemma 8.1.To estimate V11, we recall the propagation equation for trχ ′:
Ltrχ ′ + 2
t − utrχ ′ = etrχ ′ − |χ ′|2 + 2e
t − u− trα′ := ρ0
Applying Rαi to this equation gives:
LRαi trχ
′ + 2
t − uRαi trχ
′ =∑
|β|≤|α|Rβi(Ri )Z Rα−β−1
i trχ ′ + Rαi ρ0
123
On the formation of shocks for quasilinear wave equations
Again, by the estimates
|(Ri )Z | ≤ Cδ, |Tψ | ≤ Cδ−1/2
as well as Proposition 7.4, the contribution of the first term on the right handside is bounded in terms of modified energies by:
δ1/2∫ t
−r0‖Rα
i trχ′‖L2(�
ut ′ )
‖Rα+2i ψ‖L2(�
ut ′ )dt ′
≤ Cδ∫ t
−r0μ
−2b|α|+2−1m (t ′)E≤|α|+2(t
′, u)dt ′ ≤Cδμ−2b|α|+2m (t)E≤|α|+2(t, u).
(9.31)
The following terms in Rαρ0 also enjoy the above estimates:
Rαi
(|χ ′|2), (Rβi e) · (Rα−β
i trχ ′) where |β| ≤ |α| − |β|
While the other contributions from Rαi e · trχ ′, 2e
t−u and the lower order terms
in trα′ can be bounded by:
Cμ−2b|α|+2+1/2m (t)
(δ3/2 E|α|+2(t, u) + δ3/2 E≤|α|+2(t, u)
)(9.32)
in view of Lemma 7.3.Now we estimate the contribution from the principal term in trα′, which is:
dc2
dρψ0 /�Rα
i ψ0
In view of the estimates:
|ψ0| ≤ Cδ1/2, |Tψ | ≤ Cδ−1/2
this contribution is bounded by:
∫ t
−r0‖/dRα+1
i ψ0‖L2(�ut ′ )
‖Rα+2i ψ‖L2(�
ut ′ )dt ′ ≤ C
∫ t
−r0‖/dRα+1
i ψ‖2L2(�
ut ′ )dt ′
≤ C∫ t
−r0μ
−2b|α|+2−1m (t ′)E≤|α|+2(t
′, u)dt ′
123
S. Miao, P. Yu
Again, considering the“shock part” and “non-shock part” in regard to thisintegral, we obtain that it is bounded by:
C
2b|α|+2μ
−2b|α|+2m (t)E≤|α|+2(t, u) +
∫ t0
−r0μ
−2b|α|+2m (t ′)E≤|α|+2(t
′, u)dt ′
(9.33)
This completes the estimates for the spacetime integral (9.26).Next we consider the top order optical contribution of the variation
Rα′T l+1ψ , where |α′| + l + 1 = |α| + 1, which is the following spacetime
integral:
δ2l+2
∣∣∣∣∣
∫
Wtu
(Tψ) · (Rα′i T l /�μ) · (LRα′
i T l+1ψ)dt ′du′dμ/g
∣∣∣∣∣
(9.34)
Again, we rewrite the above spacetime integral as:
δ2l+2∫
Wtu
(Tψ) · (Rα′i T l /�μ) · ((L + trχ)(Rα′
i T l+1ψ))dt ′du′dμ/g
− δ2l+2∫
Wtu
(Tψ) · (Rα′i T l /�μ) · (trχ(Rα′
i T l+1ψ))dt ′du′dμ/g
which is:
δ2l+2∫
Wtu
(L + trχ)((Tψ)(Rα′
i T l /�μ)(Rα′i T l+1ψ)
)dt ′du′dμ/g
− δ2l+2∫
Wtu
(LTψ)(Rα′i T l /�μ)(Rα′
i T l+1ψ)dt ′du′dμ/g
− δ2l+2∫
Wtu
(Tψ)((L + trχ)(Rα′
i T l /�μ))(Rα′
i T l+1ψ)dt ′du′dμ/g
As before, the spacetime integral in the first line above can be written as:
δ2l+2∫
�ut
(Tψ)(Rα′i T l /�μ)(Rα′
i T l+1ψ)du′dμ/g
− δ2l+2∫
�u−r0
(Tψ)(Rα′i T l /�μ)(Rα′
i T l+1ψ)du′dμ/g
123
On the formation of shocks for quasilinear wave equations
We shall only estimate the integral on �ut , which can be written as:
−δ2l+2∫
�ut
(Tψ)(Rα′−1i T l /�μ)(Rα′+1
i T l+1ψ)du′dμ/g
− δ2l+2∫
�ut
((RiTψ) + 1
2tr(Ri )
/π
)(Rα′−1
i T l /�μ)(Rα′i T l+1ψ)du′dμ/g
:= −H ′0 − H ′
1
By the estimates:
|RiTψ | ≤ Cδ−1/2, |tr(Ri ) /π | ≤ Cδ
we see that compared to H ′0, H
′1 is a lower order term, so here we only give
the estimates for H ′0:
|H ′0| ≤ Cδ−1/2+2l+2‖Rα′−1T l /�μ‖L2(�
ut )
‖/dRα′i T l+1ψ‖L2(�
ut )
In view of Proposition 7.5, this is bounded in terms of modified energies by:
|H ′0| ≤ C
∫ t
−r0δ(μ
−b|α|+2−1/2m (t ′)
√E≤|α|+2(t
′, u)
+μ−b|α|+2m (t ′)
√E≤|α|+2(t ′, u)
)dt ′
·μ−b|α|+2−1/2m (t)
√E≤|α|+2(t, u)
Again, we need to consider the “shock part H ′S0 ” and the “non-shock part H ′N
0 ”separately.
For H ′S0 , we have:
|H ′S0 | ≤ Cμ
−b|α|+2−1/2m (t)
√E≤|α|+2(t, u)
· δ(√
E≤|α|+2(t, u)∫ t
t0μ
−b|α|+2−1/2m (t ′)dt ′
+√E≤|α|+2(t, u)
∫ t
t0μ
−b|α|+2m (t ′)dt ′
)
≤ Cδ(b|α|+2 − 1/2
)μ−2b|α|+2m (t)E≤|α|+2(t, u)
+ Cδ(b|α|+2 − 1
)μ−2b|α|+2+1/2m (t)
√E≤|α|+2(t, u)
√E≤|α|+2(t, u)
123
S. Miao, P. Yu
For H ′N0 , we use the same argument as we did in estimating (9.27). Since
μ−1m (t ′) ≤ C for t ′ ∈ [−r0, t0] with an absolute constant C , we have, by
(8.10):
∫ t0
−r0μ
−b|α|+2−1/2m (t ′)
√E≤|α|+2(t
′, u)
≤ C∫ t0
−r0μ
−b|α|+2+1/2m (t ′)
√E≤|α|+2(t
′, u)dt ′
≤ C∫ t0
−r0
√E≤|α|+2(t
′, u)dt ′ · μ−b|α|+2+1/2m (t)
The sameargument applies to the integral involving δ√E≤|α|+2(t, u), sofinally
we obtain:
|H ′N0 | ≤ Cδ
∫ t0
−r0
√E≤|α|+2(t
′, u)dt ′ · μ−2b|α|+2m (t)
√E≤|α|+2(t, u)
+C∫ t0
−r0δ
√E≤|α|+2(t ′, u)dt ′ · μ−2b|α|+2
m (t)√E≤|α|+2(t, u)
≤ Cδμ−2b|α|+2m (t)
(E≤|α|+2(t, u) +
√E≤|α|+2(t, u)
√E≤|α|+2(t, u)
)
We finally obtain the estimates for |H ′0|:
|H ′0| ≤ Cδμ
−2b|α|+2m (t)
(E≤|α|+2(t, u) +
√E≤|α|+2(t, u)
√E≤|α|+2(t, u)
)
(9.35)
Finally, we estimate the spacetime integrals:
−δ2l+2∫
Wtu
(LTψ)(Rα′i T l /�μ)(Rα′
i T l+1ψ)dt ′du′dμ/g
− δ2l+2∫
Wtu
(Tψ)
((L + 2
t − u
)(Rα′
i T l /�μ)
)(Rα′
i T l+1ψ)dt ′du′dμ/g
− δ2l+2∫
Wtu
(Tψ) · trχ ′(Rα′i T l /�μ) · (Rα′
i T l+1ψ)dt ′du′dμ/g
:= −V ′1 − V ′
2 − V ′3
By the estimates:
|Tψ | ≤ Cδ−1/2, |LTψ | ≤ Cδ−1/2, |trχ ′| ≤ Cδ
123
On the formation of shocks for quasilinear wave equations
we see that, compared to V ′1, V
′3 is a lower order term. By Lemma 7.3 and
(9.22), V ′1 is bounded by:
|V ′1| ≤ Cδ−1/2
∫ t
−r0sup�
ut ′(μ−1)δl+1‖μRα′
i T l /�μ‖L2(�ut ′ )
· δ(μ
−b|α|+2m (t ′)
√E≤|α|+2(t, u) + μ
−b|α|+2m (t ′)
√E≤|α|+2(t
′, u))dt ′
≤ C∫ t
−r0μ
−2b|α|+2−1m (t ′)
(√E≤|α|+2(t
′, u) +√E≤|α|+2(t)
)
· δ(√
E≤|α|+2(t, u) +√E≤|α|+2(t
′, u))dt ′
+C∫ t
−r0μ
−2b|α|+2m (t ′)
√∫ u
0F≤|α|+2(t, u
′)du′
×(√
E≤|α|+2(t′, u) +
√E≤|α|+2(t, u)
)dt ′
Again, we consider the “shock part” and “non-shock part” separately. Whent ′ ∈ [−r0, t0], since μ−1
m (t ′) ≤ C , the above integrals are bounded by:
Cδ∫ t0
−r0μ
−2b|α|+2m (t ′)
(E≤|α|+2(t, u) + E≤|α|+2(t
′, u))dt ′
+C∫ t0
−r0δ−1μ
−2b|α|+2m (t ′)
(∫ u
0F≤|α|+2(t
′, u′)du′)dt ′
While when t ′ ∈ [t0, t], we have the following estimates:
Cδ(E≤|α|+2(t, u) + E≤|α|+2(t, u)
) ∫ t
t0μ
−2b|α|+2−1m (t ′)dt ′
+ δ−1(∫ u
0F≤|α|+2(t, u
′)du′)∫ t
t0μ
−2b|α|+2m (t ′)dt ′
≤ Cδ
2b|α|+2μ
−2b|α|+2m (t)
(E≤|α|+2(t, u) + E≤|α|+2(t, u)
)
+ Cδ−1(2b|α|+2 − 1
)μ−2b|α|+2+1m (t)
∫ u
0F≤|α|+2(t, u
′)du′
123
S. Miao, P. Yu
Therefore V ′1 is bounded by:
Cδ
2b|α|+2μ
−2b|α|+2m (t)
(E≤|α|+2(t, u) + E≤|α|+2(t, u)
)
+ Cδ−1(2b|α|+2 − 1
)μ−2b|α|+2+1m (t)
∫ u
0F≤|α|+2(t, u
′)du′
+Cδ∫ t0
−r0μ
−2b|α|+2m (t ′)
(E≤|α|+2(t, u) + E≤|α|+2(t
′, u))dt ′
+C∫ t0
−r0δ−1μ
−2b|α|+2m (t ′)
(∫ u
0F≤|α|+2(t
′, u′)du′)dt ′ (9.36)
Now we estimate V ′2. First we write V
′2 as:
δ2l+2∫
Wtu
(Tψ)
(Ri
(L + 2
t − u
)Rα′−1i T l /�μ
)(Rα′
i T l+1ψ)dt ′du′dμ/g
+ δ2l+2∫
Wtu
(Tψ)((Ri )Z Rα′−1i T l /�μ)(Rα′
i T l+1ψ)dt ′du′dμ/g := V ′21 + V ′
22
By the estimate:
|(Ri )Z | ≤ Cδ
V ′22 has the identical structurewithV
′3, therefore is a lower order termcompared
to V ′1.
While for V ′21, integrating by parts, we have:
V ′21 = −δ2l+2
∫
Wtu
((RiTψ) + 1
2tr(Ri )
/π
)((L + 2
t − u
)Rα′−1i T l /�μ
)
× (Rα′i T l+1ψ)dt ′du′dμ/g
− δ2l+2∫
Wtu
(Tψ)
((L + 2
t − u
)Rα′−1i T l /�μ
)
× (Rα′+1i T l+1ψ)dt ′du′dμ/g := −V ′
211 − V ′212
By the estimates:
|Tψ | ≤ Cδ−1/2,
∣∣∣∣RiTψ + 1
2(Ri )tr /π
∣∣∣∣ ≤ Cδ−1/2
123
On the formation of shocks for quasilinear wave equations
as well as Lemma 7.3, V ′211 is a lower order term compared to V ′
212. To estimateV ′212, we first consider:
δ2l+2∫
Wtu
2
t − u(Tψ)(Rα′−1
i T l /�μ)(Rα′+1i T l+1ψ)dt ′du′dμ/g (9.37)
By Proposition 7.5, this is bounded by:
C∫ t
−r0δ(μ
−b|α|+2−1/2m (t ′)
√E≤|α|+2(t
′, u) + μ−b|α|+2m (t ′)
√E≤|α|+2(t ′, u)
)
·(μ
−b|α|+2−1/2m (t ′)
√E≤|α|+2(t
′, u))dt ′
Again,we consider the “shock part” and the “non-shock part” separately.Whent ′ ∈ [−r0, t0], the above integrals are bounded by:
C∫ t0
−r0μ
−2b|α|+2m (t ′)
(δ2 E≤|α|+2(t
′, u) + E≤|α|+2(t′, u)
)dt ′
While when t ′ ∈ [t0, t], the above integrals are bounded by:
C E≤|α|+2(t, u)∫ t
t0μ
−2b|α|+2−1m (t ′)dt ′
+Cδ2 E≤|α|+2(t, u)∫ t
t0μ
−2b|α|+2m (t ′)dt ′
≤ C
2b|α|+2μ
−2b|α|+2m (t)E≤|α|+2(t, u)
+ C(2b|α|+2 − 1
)μ−2b|α|+2+1m (t)δ2 E≤|α|+2(t, u)
Therefore the spacetime integral (9.37) is bounded by:
C
2b|α|+2μ
−2b|α|+2m (t)E≤|α|+2(t, u)
+ C(2b|α|+2 − 1
)μ−2b|α|+2+1m (t)δ2 E≤|α|+2(t, u)
+C∫ t0
−r0μ
−2b|α|+2m (t ′)
(δ2 E≤|α|+2(t
′, u) + E≤|α|+2(t′, u)
)dt ′ (9.38)
123
S. Miao, P. Yu
We end this section by estimating the spacetime integral:
δ2l+2∫
Wtu
(Tψ)(LRα′−1i T l /�μ)(Rα′+1
i T l+1ψ)dt ′du′dμ/g (9.39)
which is bounded by:
Cδ−1/2∫ t
−r0δl+1‖LRα′−1
i T l /�μ‖L2(�ut ′ )δl+1‖Rα′+1
i T l+1ψ‖L2(�ut ′ )dt ′
By the propagation equation:
Lμ = m + μe
we have:
LRα′−1i T l /�μ = −dc2
dρψ0T Rα′−1
i T l /�ψ0 + μψ0LRα′−1i T l /�ψ0 + l.o.t.
Here the lower order terms l.o.t. can be bounded in the same fashion as(9.38). Now we are going to bound δl+1‖T Rα′−1
i T l /�ψ0‖L2(�ut ′ )
in terms of√E≤|α|+2(t ′) and δl+1‖μLRα′−1
i T l /�ψ0‖L2(�ut ′ )
in terms of√E≤|α|+2(t
′), sothe latter is a lower order term with respect to the behavior of δ. We onlyestimate the contribution from the former one.
In view of the estimate:
|ψ0| ≤ Cδ1/2
This contribution to (9.39) is bounded by:
C∫ t
−r0δ
√E≤|α|+2(t ′, u)μ−1/2
m (t ′)√E≤|α|+2(t
′, u)dt ′
≤ C∫ t
−r0μ
−2a−1/2m (t ′)δ
√E≤|α|+2(t ′, u)
√E≤|α|+2(t
′, u)dt ′
Again, considering the “shock part” and “non-shock part” separately, we havethe finally estimates for (9.39):
123
On the formation of shocks for quasilinear wave equations
C∫ t0
−r0μ
−2b|α|+2m (t ′)
(δ2 E≤|α|+2(t
′, u) + E≤|α|+2(t′, u)
)dt ′
+ C(2b|α|+2 − 1/2
)μ−2b|α|+2+1/2m (t)
(δ2 E≤|α|+2(t, u) + E≤|α|+2(t, u)
)
(9.40)
This completes the error estimates for the top order optical terms. As wementioned at the beginning of this subsection, although we only consideredthe variations Rα+1
i ψ and Rα′i T l+1ψ with |α′|+l = |α|, all the estimates in this
subsection are also true for the variations Zα+1i ψ and Zα′
i T l+1ψ respectively.Here Zi is either Ri or Q.
10 Top order energy estimates
With the estimates for the contributions from top order optical terms as wellas lower order optical terms, we are ready to complete the top order energyestimates, namely, the energy estimates for the variations of order up to |α|+2.As we have pointed out, we allow the top order energies to blow up as shocksform.So in this section,we shall prove that themodified energies E≤|α|+2(t, u),E≤|α|+2(t, u) and F≤|α|+2(t, u), F≤|α|+2(t, u) are bounded by initial data.Therefore we obtain a rate for the possible blow up of the top order energies.
10.1 Estimates associated to K1
We start with the energy inequality for Zα′+1i ψ as we obtained in Sect. 6. Here
Zi is any one of Ri Q and T .
∑
|α′|≤|α|δ2l
′(E[Zα′+1
i ψ](t, u) + F[Zα′+1i ψ](t, u) + K [Zα′+1
i ψ](t, u))
≤ C∑
|α′|≤|α|δ2l
′E[Zα′+1
i ψ](−r0, u) + C∑
|α′|≤|α|δ2l
′∫
Wtu
c−2 Q1,|α′|+2
where l ′ is the number of T s’ appearing in the string of Zα′i . In the spacetime
integral∫Wt
uc−2 Q1,α′+2 we have the contributions from the deformation tensor
of K1, which have been investigated in Sect. 6. Actually, if we choose Ntop to
be large enough, then we can bound ‖/dZβi μ‖L∞(�
ut )in terms of initial data by
using the same argument as in Sect. 4.2 for |β| ≤ N∞ + 1.Another contribution of the spacetime integral
∫Wt
uc−2 Q1,|α′|+2 comes from
∫Wt
u
1c ρ|α′|+2 · LZα′+1
i ψ , namely, the deformation tensor of commutators,
123
S. Miao, P. Yu
which has been studied intensively in the last section. We first consider thelower order optical contributions, which are bounded by (see (9.14)):
C∫ t
−r0δ2E≤|α|+2(t
′, u)dt ′ + Cδ−1/2∫ u
0F≤|α|+2(t, u
′)du′
+Cδ1/2K≤|α|+2(t, u) ≤ Cμ−2b|α|+2m (t)
(∫ t
−r0δ2 E≤|α|+2(t
′, u)dt ′
+ δ−1/2∫ u
0F≤|α|+2(t, u
′)du′ + δ1/2 K≤|α|+2(t, u)
). (10.1)
Here we define the following non-decreasing quantity in t :
K≤|α|+2(t, u) := supt ′∈[−r0,t]
{μ2b|α|+2m (t ′)K≤|α|+2(t
′, u)}
By (9.28)–(9.33), (9.35), (9.36), (9.38) and (9.40) the contribution of toporder optical terms are bounded as (provided that δ is sufficiently small):
C(b|α|+2 − 1/2
)μ−2b|α|+2m (t)E≤|α|+2(t, u)
+Cεμ−2b|α|+2m (t)
∫ t0
−r0E≤|α|+2(t
′, u)dt ′
+ εμ−2b|α|+2m (t)E≤|α|+2(t, u)
+ Cδ−1(2b|α|+2 − 1
)μ−2b|α|+2+1m (t)
∫ u
0F≤|α|+2(t, u
′)du′
+Cδ∫ t0
−r0μ
−2b|α|+2m (t ′)
(E≤|α|+2(t
′, u) + δ−2∫ u
0F≤|α|+2(t
′, u′)du′)dt ′
+Cδμ−2b|α|+2m (t)E≤|α|+2(t, u)
+Cμ−2b|α|+2+1/2m (t)
(δ3/2 E≤|α|+2(t, u) + δ3/2 E≤|α|+2(t, u)
)
+ C
2b|α|+2μ
−2b|α|+2m (t)E≤|α|+2(t, u) + Cδ
2b|α|+2μ
−2b|α|+2m (t)E≤|α|+2(t, u)
(10.2)
Substituting these contributions into the energy inequality, and use the factthat μm(t) ≤ 1, we obtain:
123
On the formation of shocks for quasilinear wave equations
Now the right hand side of the above inequality is non-decreasing in t , so theabove inequality is also valid if we replace “t” by any t ′ ∈ [−r0, t] on the lefthand side:
μ2b|α|+2m (t ′)
∑
|α′|≤|α|
(E[Zα′+1
i ψ](t ′, u) + F[Zα′+1i ψ](t ′, u)
+K [Zα′+1i ψ](t ′, u)
)
≤ C∑
|α′|≤|α|δ2l
′E[Zα′+1
i ψ](−r0, u) + C(b|α|+2 − 1/2
) E≤|α|+2(t, u)
+C∫ t
−r0E≤|α|+2(t
′, u)dt ′ + ε E≤|α|+2(t, u)
+ Cδ−1(2b|α|+2 − 1
)∫ u
0F≤|α|+2(t, u
′)du′
+Cδ∫ t
−r0
(E≤|α|+2(t
′, u) + δ−2∫ u
0F≤|α|+2(t
′, u′)du′)dt ′
+Cδ E≤|α|+2(t, u)
+C(δ3/2 E≤|α|+2(t, u) + δ3/2 E≤|α|+2(t, u)
)+ C
2b|α|+2E≤|α|+2(t, u)
+ Cδ
2b|α|+2E≤|α|+2(t, u) + Cδ K≤|α|+2(t, u)
123
S. Miao, P. Yu
For each term in the sum on the left hand side of above inequality, we keep iton the left hand side and drop all the other terms. Then taking supremum ofthe term we kept with respect to t ′ ∈ [−r0, t]. Repeat this process for all theterms on the left hand side, we finally obtain:
The control on the boxed term relies on Remark 9.2—since Cb|α|+2
is suitablysmall, the boxed term can be absorbed by the left hand side. So if we chooseε and δ small enough, we obtain:
E≤|α|+2(t, u) + F≤|α|+2(t, u) + K≤|α|+2(t, u)
≤ C∑
|α′|≤|α|δ2l
′E[Zα+1
i ψ](−r0, u) + Cε
∫ t
−r0E≤|α|+2(t
′, u)dt ′
+Cδ−1∫ u
0F≤|α|+2(t, u
′)du′ + Cδ∫ t
−r0E≤|α|+2(t
′, u)dt ′
+Cδ3/2 E≤|α|+2(t, u) (10.3)
Now we only keep E≤|α|+2(t, u) on the left hand side of (10.3)
E≤|α|+2(t, u) ≤ C∑
|α′|≤|α|δ2l
′E[Zα′+1
i ψ](−r0, u)
+Cε
∫ t
−r0E≤|α|+2(t
′, u)dt ′
123
On the formation of shocks for quasilinear wave equations
+Cδ−1∫ u
0F≤|α|+2(t, u
′)du′ + Cδ∫ t
−r0E≤|α|+2(t
′, u)dt ′
+Cδ3/2 E≤|α|+2(t, u)
Then by using Gronwall we have:
E≤|α|+2(t, u) ≤ C∑
|α′|≤|α|δ2l
′E[Zα′+1
i ψ](−r0, u) + Cδ−1
∫ u
0F≤|α|+2(t, u
′)du′
+Cδ∫ t
−r0E≤|α|+2(t
′, u)dt ′ + Cδ3/2 E≤|α|+2(t, u)
(10.4)
Keeping only F≤|α|+2(t, u) on the left hand side of (10.3) and substituting theabove estimates for E≤|α|+2(t, u) gives us:
F≤|α|+2(t, u) ≤ C∑
|α′|≤|α|δ2l
′E[Zα′+1
i ψ](−r0, u)
+Cδ−1∫ u
0F≤|α|+2(t, u
′)du′
+Cδ∫ t
−r0E≤|α|+2(t
′, u)dt ′ + Cδ3/2 E≤|α|+2(t, u)
Then again by using Gronwall we obtain:
F≤|α|+2(t, u) ≤ C∑
|α′|≤|α|δ2l
′E[Zα′+1
i ψ](−r0, u)
+Cδ∫ t
−r0E≤|α|+2(t
′, u)dt ′ + Cδ3/2 E≤|α|+2(t, u)
(10.5)
Note that since 0 ≤ u ≤ δ, all the constants C on the above do not dependon δ. Since the right hand side of (10.5) is increasing in u, substituting this in(10.4) gives us:
123
S. Miao, P. Yu
E≤|α|+2(t, u)≤C∑
|α′|≤|α|δ2l
′E[Zα′+1
i ψ](−r0, u)+Cδ∫ t
−r0E≤|α|+2(t
′, u)dt ′
+Cδ3/2 E≤|α|+2(t, u) (10.6)
Substituting (10.5) and (10.6) in (10.3) gives us:
E≤|α|+2(t, u) + F≤|α|+2(t, u) + K≤|α|+2(t, u)
≤ C∑
|α′|≤|α|δ2l
′E[Zα′+1
i ψ](−r0, u)
+Cδ∫ t
−r0E≤|α|+2(t
′, u)dt ′ + Cδ3/2 E≤|α|+2(t, u) (10.7)
This completes the top order energy estimates associated to K1.
10.2 Estimates associated to K0
Now we turn to the top order energy estimates for K0.We first start with the energy identity for Zα′+1
i ψ , where Zi is any one ofRi , Q and T :
∑
|α′|≤|α|δ2l
′ (E[Zα′+1
i ψ](t, u) + F[Zα′+1i ψ](t, u)
)
≤ C∑
|α′|≤|α|δ2l
′E[Zα′+1
i ψ](−r0, u) + C∑
|α′|≤|α|
∫
Wtu
c−2 Q0,|α′|+2
Again, l ′ is the number of T s’ in the string of Zα′+1i .
In the spacetime integral∫Wt
uc−2 Q0,α′+2 we have the contributions from the
deformation tensor of K0, which have been investigated in Sect. 6 and also thecontribution of the spacetime integral from
∫Wt
u
1c ρ|α′|+2 · LZα′+1
i ψ , namely,
the deformation tensor of commutators, which has been studied intensively inthe last section. We first consider the lower order optical contributions, whichare bounded by (see (9.15) and (10.7) and provided that δ is sufficiently small):
C∫ t
−r0δ1/2E≤|α|+2(t
′, u)dt ′ + Cδ1/2K≤|α|+2(t, u)
+ δ−1/2∫ u
0F≤|α|+2(t, u
′)du′
123
On the formation of shocks for quasilinear wave equations
≤ CE≤|α|+2(−r0, u) + Cδ1/2μ−2b|α|+2m (t)
∫ t
−r0E≤|α|+2(t
′, u)dt ′.
(10.8)
By (9.18), (9.19), (9.23), (9.24) and (10.7), the top order optical contribu-tions are bounded by (provided that δ is sufficiently small and b|α|+2 is largeenough):
∫ t
−r0μ
−2b|α|+2m (t ′)E≤|α|+2(t
′, u)dt ′ + δμ−2b|α|+2m (t)E≤|α|+2(t, u)
+ C
2b|α|+2μ
−2b|α|+2m (t)E≤|α|+2(t, u)
+ C(2b|α|+2 − 1
)μ−2b|α|+2+1m (t)
(∫ t
−r0E≤|α|+2(t
′, u)dt ′ + E≤|α|+2(t, u)
)
+Cδ−1+2l ′ ∑
|α′|≤|α|E[Zα′+1
i ψ](−r0, u) (10.9)
Substituting (10.8) and (10.9) into the energy inequality, and use the fact thatμm(t) ≤ 1, we obtain:
∑
|α′|≤|α|μ2b|α|+2m (t ′)δ2l ′
(E[Zα′+1
i ψ](t, u) + F[Zα′+1i ψ](t, u)
)
≤ C∑
|α′|≤|α|δ2l
′E[Zα′+1
i ψ](−r0, u)+C∑
|α′|≤|α|δ−1+2l ′E[Zα′+1
i ψ](−r0, u)
+C∫ t
−r0E≤|α|+2(t
′, u)dt ′ + δ E≤|α|+2(t, u) + C
2b|α|+2E≤|α|+2(t, u)
Since the right hand side of the above is non-decreasing in t , the inequality istrue if we replace t by any t ′ ∈ [−r0, t] on the left hand side:∑
|α′|≤|α|δ2l
′μ2b|α|+2m (t ′)
(E[Zα′+1
i ψ](t ′, u) + F[Zα′+1i ψ](t ′, u)
)
≤ C∑
|α′|≤|α|δ2l
′E[Zα′+1
i ψ](−r0, u) + Cδ−1+2l ′ ∑
|α′|≤|α|E[Zα′+1
i ψ](−r0, u)
≤ C∫ t
−r0E≤|α|+2(t
′, u)dt ′ + δ E≤|α|+2(t, u) + C
2b|α|+2E≤|α|+2(t, u)
123
S. Miao, P. Yu
As before, taking supremum on the left hand side with respect to t ′ ∈ [−r0, t],we obtain:
E≤|α|+2(t, u) + F≤|α|+2(t, u)
≤ C∑
|α′|≤|α|δ2l
′E[Zα′+1
i ψ](−r0, u) + Cδ−1+2l ′ ∑
|α′|≤|α|E[Zα′+1
i ψ](−r0, u)
+C∫ t
−r0E≤|α|+2(t
′, u)dt ′ + δ E≤|α|+2(t) + C
2b|α|+2E≤|α|+2(t, u)
Similarly, the control on the boxed term relies on Remark 9.2—since Cb|α|+2
is suitably small, the boxed term can be absorbed by the left hand side. Bychoosing δ sufficiently small, we have:
E≤|α|+2(t) + F≤|α|+2(t, u) ≤ C∑
|α′|≤|α|δ2l
′E[Zα′+1
i ψ](−r0, u)
+Cδ−1+2l ′ ∑
|α′|≤|α|E[Zα′+1
i ψ](−r0, u)
+C∫ t
−r0E≤|α|+2(t
′, u)dt ′
Then keeping only E≤|α|+2(t) on the left hand side gives us:
E≤|α|+2(t) ≤ C∑
|α′|≤|α|δ2l
′E[Zα′+1
i ψ](−r0, u)
+Cδ−1+2l ′ ∑
|α′|≤|α|E[Zα′+1
i ψ](−r0, u)
+C∫ t
−r0E≤|α|+2(t
′, u)dt ′
Then using Gronwall, we have:
E≤|α|+2(t) ≤ C∑
|α′|≤|α|δ2l
′E[Zα′+1
i ψ](−r0, u)
+Cδ−1+2l ′ ∑
|α′|≤|α|E[Zα′+1
i ψ](−r0, u)
123
On the formation of shocks for quasilinear wave equations
Therefore
E≤|α|+2(t, u) + F≤|α|+2(t, u)
≤ C∑
|α′|≤|α|δ2l
′E[Zα′+1
i ψ](−r0, u) + Cδ−1+2l ′
∑
|α′|≤|α|E[Zα′+1
i ψ](−r0, u)
(10.10)
Now we substitute this to (10.7) for E≤|α|+2(t):
E≤|α|+2(t, u) + F≤|α|+2(t, u) + K≤|α|+2(t, u)
≤ C∑
|α′|≤|α|δ2l
′E[Zα′+1
i ψ](−r0, u) + C∑
|α′|≤|α|δ2l
′+1E[Zα′+1i ψ](−r0, u)
(10.11)
If we denote:
Du|α|+2 :=
∑
|α′|+l ′≤|α|δ2l
′E[Zα′+1
i ψ](−r0, u)
+ δ−1+2l ′ ∑
|α′|+l ′≤|α|E[Zα′+1
i ψ](−r0, u)
+ δ2l+2‖Fα,l‖L2(�u−r0
)
+∑
|α′|+l ′≤|α|+1
δ2l′‖Zα′
i T l ′μ‖L2(�u−r0
), Zi = Ri , Q.
we can write the final top order energy estimates as:
In the previous section, we have shown that the modified energies E≤|α|+2(t),E≤|α|+2(t) for the top order variations are bounded by the initial energies
Du|α|+2. According to the definition, the modified energies go to zero when
123
S. Miao, P. Yu
μm(t) goes to zero. This means the energy estimates obtained in the lastsection are not sufficient for us to close the argument when shock forms.However, based on those estimates, we shall show in this section, that if theorder of derivative decreases, the power of μm(t) needed in the definition ofmodified energies also decreases. The key point is that after several steps,this power could be zero and finally we can bound the energies without anyweights.
11.1 Next-to-top order error estimates
We first investigate the estimates associated to K1. To improve the energyestimates for the next-to-the-top variations, we consider the spacetime integral(Keep in mind that the top order quantities are of order |α| + 2):
∫
Wtu
|Tψ ||Zαi trχ
′||LZαi ψ |dt ′du′dμg
≤ Cδ−1/2∫
Wtu
|Zαi trχ
′||LZαi ψ |dt ′du′dμg
≤ Cδ−1/2
(∫
Wtu
|Zαi trχ
′|2dt ′du′dμ/g
)1/2
·(∫
Wtu
|LZαi ψ |2dt ′du′dμ/g
)1/2
≤ Cδ−1/2(∫ t
−r0‖Zα
i trχ′‖2
L2(�ut ′ )dt ′)1/2
·(∫ u
0F[Zα
i ψ](t, u′)du′)1/2
(11.1)
Throughout this subsection, Zi is either Ri or Q.By Proposition 7.4:
‖Zαi trχ
′‖L2(�ut )
≤ Cδ1/2∫ t
−r0μ
−1/2m (t ′)
√E |α|+2(t
′, u)dt ′
≤ Cδ1/2∫ t
−r0μ
−1/2−b|α|+2m (t ′)
√E≤|α|+2(t
′, u)dt ′
≤ Cδ1/2√E≤|α|+2(t, u)
∫ t
−r0μ
−b|α|+2−1/2m (t ′)dt ′
≤ Cδ1/2μ−b|α|+2+1/2m (t)
√E≤|α|+2(t, u)
Then by the top order energy estimates obtained in the last section, the integralin the first factor of (11.1) is bounded by:
123
On the formation of shocks for quasilinear wave equations
Cδ∫ t
−r0μ
−2b|α|+2+1m (t ′)E≤|α|+2(t
′, u)dt ′
≤ Cδ E≤|α|+2(t, u)∫ t
−r0μ
−2b|α|+2+1m (t ′)dt ′
≤ Cδ2μ−2b|α|+2+2m (t)Du
|α|+2
On the other hand, the second factor in (11.1) is bounded by:
∫ u
0F[Zα
i ψ](t, u′)du′
≤ μ−2b|α|+1m (t)
∫ u
0sup
t ′∈[−r0,t]
{μ2b|α|+1m (t ′)F[Zα
i ψ](t ′, u′)}du′
where b|α|+1 = b|α|+2 − 1. Therefore (11.1) is bounded by:
Cδ1/2μ−2b|α|+1m (t)
√Du
|α|+2
√∫ u
0F≤|α|+1(t, u
′)du′
≤ Cδ2μ−2b|α|+1m (t)Du
|α|+2 + Cδ−1μ−2b|α|+1m (t)
∫ u
0F≤|α|+1(t, u
′)du′
(11.2)
Next we consider the spacetime integral:
δ2l′+2∫
Wtu
|Tψ ||Zα′i T l ′ /�μ||LZα′
i T l ′+1ψ |dt ′du′dμ/g
≤ Cδ−1/2(∫ t
−r0δl
′+1‖Zα′i T l ′ /�μ‖2
L2(�ut ′ )
)1/2
×(∫ u
0δl
′+1F[Zα′i T l ′+1ψ](t, u′)du′
)1/2
(11.3)
with |α′| + l ′ ≤ |α| − 1.By Proposition 7.5:
δl′+1‖Zα′
i T l ′ /�μ‖L2(�ut )
≤ Cδ1/2∫ t
−r0
√E≤|α|+2(t ′, u) + μ
−1/2m (t ′)
√E≤|α|+2(t
′, u)dt ′
≤ Cδ1/2∫ t
−r0μ
−b|α|+2m (t ′)
√E≤|α|+2(t ′, u)
123
S. Miao, P. Yu
+μ−b|α|+2−1/2m (t ′)
√E≤|α|+2(t
′, u)dt ′
≤ Cδ1/2(√
E≤|α|+2(t, u)∫ t
−r0μ
−b|α|+2m (t ′)dt ′
+√E≤|α|+2(t, u)
∫ t
−r0μ
−b|α|+2−1/2m (t ′)dt ′
)
≤ Cδ1/2μ−b|α|+2+1/2m (t)
√E≤|α|+2(t, u)
+Cδ1/2μ−b|α|+2+1m (t)
√E≤|α|+2(t, u)
Then by the top order energy estimates obtained in the last section, the integralin the first factor of (11.3) is bounded by (μm(t) ≤ 1):
Cδ∫ t
−r0μ
−2b|α|+2+1m (t ′)
(E≤|α|+2(t
′, u) + E≤|α|+2(t′, u)
)dt ′
≤ Cδ(E≤|α|+2(t) + E≤|α|+2(t, u)
) ∫ t
−r0μ
−2b|α|+2+1m (t ′)dt ′
≤ Cδμ−2b|α+2+2m (t)
(E≤|α|+2(t, u) + E≤|α|+2(t, u)
)
≤ Cδμ−2b|α+2+2m (t)Du
|α|+2
Then again, with b|α|+1 = b|α|+2−1, the spacetime integral (11.3) is boundedby:
Cμ−2b|α|+1m (t)
√Du
|α|+2
√∫ u
0F≤|α|+1(t, u
′)du′
≤ Cδμ−2b|α|+1m (t)Du
|α|+2 + Cδ−1μ−2b|α|+1m (t)
∫ u
0F≤|α|+1(t, u
′)du′
(11.4)
We proceed to consider the spacetime error integral associated to K0. Wefirst consider the spacetime integral:
∫
Wtu
|Tψ ||Zαi trχ
′||LZαi ψ |dt ′du′dμ/g
≤ Cδ−1/2∫ t
−r0‖Zα
i trχ′‖L2(�
ut ′ )
‖LZαi ψ‖L2(�
ut ′ )dt ′
123
On the formation of shocks for quasilinear wave equations
Substituting the estimates:
‖Zαi trχ
′‖L2(�ut ′ )
≤ Cδ1/2μ−b|α|+2+1/2m (t ′)
√E≤|α|+2(t
′, u)
≤ Cδμ−b|α|+2+1/2m (t)
√Du
|α|+2,
‖LZαi ψ‖L2(�
ut ′ )
≤ Cμ−b|α|+1m (t ′)
√E≤|α|+1(t ′, u)
with b|α|+1 = b|α|+2−1, and using the fact that E≤|α|+1(t) are non-decreasingin t , we see that the spacetime integral is bounded by (μm(t) ≤ 1):
Cδ1/2√Du
|α|+2
√E≤|α|+1(t, u)
∫ t
−r0μ
−2b|α|+1−1/2m (t ′)dt ′
≤ Cδ1/2μ−2b|α|+1+1/2m (t)
√Du
|α|+2
√E≤|α|+1(t, u)
≤ Cμ−2b|α|+1m (t)Du
|α|+2 + Cδμ−2b|α|+1m (t)E≤|α|+1(t, u). (11.5)
Finally, we consider the spacetime integral:
δ2l′+2∫
Wtu
|Zα′i T l ′ /�μ||Tψ ||LZα′
i T l ′+1ψ |dt ′du′dμ/g
≤ Cδ2l′+2−1/2
∫ t
−r0‖Zα′
i T l ′ /�μ‖L2(�ut ′ )
‖LZα′i T l ′+1ψ‖L2(�
ut ′ )dt ′ (11.6)
for |α′| + l ′ ≤ |α| − 1. Again, substituting the estimates (μm(t) ≤ 1):
δl′+1‖Zα′
i T l ′ /�μ‖L2(�ut ′ )
≤ Cδ1/2μ−b|α|+2+1/2m (t ′)
(√E≤|α|+2(t ′, u) +
√E≤|α|+2(t
′, u))
with b|α|+1 = b|α|+2−1, the same argument implies that the spacetime integralis bounded by:
C√Du
|α|+2
√E≤|α|+1(t, u)
∫ t
−r0μ
−2b|α|+1−1/2m (t ′)dt ′
≤ Cμ−2b|α|+1+1/2m (t)
√Du
|α|+2,l ′√E≤|α|+1(t, u)
≤ Cεμ−2b|α|+1m (t)Du
|α|+2 + ε E≤|α|+1(t, u) (11.7)
Here ε is a small absolute positive constant.
123
S. Miao, P. Yu
11.2 Energy estimates for next-to-top order
Throughout this subsection Zi could be Ri , Q and T . Now we consider theother contributions from the spacetime error integrals associated to K1. Forthe variations Zα′
i ψ where |α′| ≤ |α|, the other contributions are bounded by(see (9.14)):
Cδ2∫ t
−r0E≤|α|+1(t
′, u)dt ′ + Cδ−1/2∫ u
0F≤|α|+1(t, u
′)du′
+Cδ1/2K≤|α|+1(t, u)
≤ Cδ2∫ t
−r0μ
−2b|α|+1m (t ′, u)E≤|α|+1(t
′, u)dt ′
+Cδ−1/2∫ u
0μ
−2b|α|+1m (t)F≤|α|+1(t, u
′)du′
+Cδ1/2μ−2b|α|+1m (t)K≤|α|+1(t, u) (11.8)
In view of (11.2), (11.4), (11.8) and multiplying μ2b|α|+1m (t) on both sides of
the energy inequality associated to K1 for Zα′i ψ with |α′| ≤ |α| give us:
∑
|α′|≤|α|μ2b|α|+1m (t)δ2l
′(E[Zα′
i ψ](t, u) + F[Zα′i ψ](t, u) + K [Zα′
i ψ](t, u))
≤ C∑
|α′|≤|α|δ2l
′E[Zα′
i ψ](−r0, u) + Cδ−1∫ u
0F≤|α|+1(t, u
′)du′
+Cδ2∫ t
−r0E≤|α|+1(t
′, u)dt ′ + Cδ1/2 K≤|α|+1(t, u) + CδDu|α|+2
Since the right hand side of the above is non-decreasing in t , the above inequal-ity is still true if we substitute t by any t ′ ∈ [−r0, t] on the left hand side:
∑
|α′|≤|α|μ2b|α|+1m (t ′)δ2l ′
(E[Zα′
i ψ](t ′, u) + F[Zα′i ψ](t ′, u) + K [Zα′
i ψ](t ′, u))
≤ C∑
|α′|≤|α|δ2l
′E[Zα′
i ψ](−r0, u) + Cδ−1∫ u
0F≤|α|+1(t, u
′)du′
+Cδ2∫ t
−r0E≤|α|+1(t
′, u)dt ′ + CδDu|α|+2 + Cδ1/2 K≤|α|+1(t, u)
123
On the formation of shocks for quasilinear wave equations
As in the previous section, taking the supremum with respect to t ′ ∈ [−r0, t]we obtain:
On the formation of shocks for quasilinear wave equations
in the role of b|α|+2 and b|α|+1 respectively, the argument beginning in theparagraph containing (11.1) and concluding with (11.12) being step 0. Thenth step is exactly the same as the 0th step as above, as long as b|α|+1−n > 0,that is, as long as n ≤ [b|α|+2] − 1. Here we choose b|α|+2 as:
b|α|+2 = [b|α|+2] + 3
4
where [b|α|+2] is the integer part of b|α|+2. For each of such n, we need toestimate the integrals:
∫ t
−r0μ
−b|α|+2−n−1/2m (t ′)dt ′,
∫ t
−r0μ
−2b|α|+2−n+1m (t ′)dt ′
As in the last section, we consider two different cases: t ′ ∈ [−r0, t0] andt ′ ∈ [t0, t], where μm(t0) = 1
10 . If t′ ∈ [−r0, t0], we have:
∫ t0
−r0μ
−b|α|+2−n−1/2m (t ′)dt ′ ≤C
∫ t0
−r0μ
−b|α|+2−n+1/2m (t ′)dt ′ ≤Cμ
−b|α|+2−n+1/2m (t)
∫ t0
−r0μ
−2b|α|+2−n+1m (t ′)dt ′ ≤ C
∫ t0
−r0μ
−2b|α|+2−n+2m (t ′)dt ′ ≤ Cμ
−2b|α|+2−n+2m (t)
Here we have used the fact that μm(t ′) ≥ 110 for t ′ ∈ [−r0, t0]. In regard to
the estimate for t ′ ∈ [t0, t], since
b|α|+2−n = [b|α|+2−n] + 3
4≥ 1 + 3
4= 7
4,
by Lemma 8.1, we have:
∫ t
t0μ
−b|α|+2−n−1/2m (t ′)dt ′ ≤ Cμ
−b|α|+2−n+1/2m (t)
∫ t
t0μ
−2b|α|+2−n+1m (t ′)dt ′ ≤ Cμ
−2b|α|+2−n+2m (t)
So indeed, we can repeat the process of 0th step for n = 1, . . . , [b|α|+2] − 1.Therefore we have the following estimates:
These are the desired estimates, because from the definitions:
E≤|α|+1−[b|α|+2](t, u) := supt ′∈[−r0,t]
{E≤|α|+1−[b|α|+2](t′, u)}
E≤|α|+1−[b|α|+2](t, u) := supt ′∈[−r0,t]
{E≤|α|+1−[b|α|+2](t ′, u)}
F≤|α|+1−[b|α|+2](t, u) := supt ′∈[−r0,t]
{F≤|α|+1−[b|α|+2](t′, u)}
F≤|α|+1−[b|α|+2](t, u) := supt ′∈[−r0,t]
{F≤|α|+1−[b|α|+2](t ′, u)}
K≤|α|+1−[b|α|+2](t, u) := supt ′∈[−r0,t]
{K≤|α|+1−[b|α|+2](t ′, u)}
(11.15)
the weight μm(t ′) has been eliminated.
12 Completion of proof
Let us define:
S2[φ] :=∫
St,u
(|φ|2 + |Ri1φ|2 + |Ri1Ri2φ|2
)dμ/g
And also let us denote by Sn(t, u) the integral on St,u (with respect to dμ/g) of
the sum of the square of all the variation ψ = δl′Zα′i ψγ up to order |α| + 1−
[b|α|+2], where l ′ is the number of T ’s in the string of Zα′i and γ = 0, 1, 2, 3.
Then by Lemma 7.3 we have:
S|α|−[b|α|+2](t, u) ≤ Cδ(E|α|+1−[b|α|+2](t) + E |α|+1−[b|α|+2](t, u)
)
for all (t, u) ∈ [−2, t∗) × [0, δ].
123
S. Miao, P. Yu
Hence, in view of (11.14) and (11.15),
S|α|−[b|α|+2](t, u) ≤ CδDu|α|+2 for all (t, u) ∈ [−2, t∗) × [0, δ]. (12.1)
Then for any variations ψ of order up to |α| − 2 − [b|α|+2] we have:S2[ψ] ≤ S|α|−[b|α|+2](t, u) (12.2)
Then by the Sobolev inequality introduced in (3.41), (12.1) and (12.2), wehave:
δl′supSt,u
|Zα′i ψα| = sup
St,u|ψ | ≤ Cδ1/2
√Du
|α|+2 ≤ C0δ1/2 (12.3)
whereC0 depends on the initial energyDu|α|+2, the constant in the isoperimetric
inequality and the constant in Lemma 7.3 as well as the constants in (11.14),which are absolute constants.
If we choose |α| large enough such that[ |α| + 1
2
]+ 3 ≤ |α| − 2 − [b|α|+2]
then (12.3) recovers the bootstrap assumption (B.1) for (t, u) ∈ [−2, t∗) ×[0, δ].
To complete the proof of Theorem 3.1, it remains to show that the smoothsolution exists for t ∈ [−2, s∗), i.e. t∗ = s∗. More precisely, we will provethat either μm(t∗) = 0 if shock forms before t = −1 or otherwise t∗ = −1.
If t∗ < s∗, then μ would be positive on �δt∗ . In particular μ has a positive
lower bound on �δt∗ . Therefore by Remark 2.4, the Jacobian � of the trans-
formation from optical coordinates to rectangular coordinates has a positivelower bound on�δ
t∗ . This implies that the inverse transformation from rectan-gular coordinates to optical coordinates is regular. On the other hand, in thecourse of recovering bootstrap assumption we have proved that all the deriv-atives of the first order variations ψα extend smoothly in optical coordinatesto �δ
t∗ . Since the inverse transformation is regular, ψα also extend smoothlyto �δ
t∗ in rectangular coordinates. Once ψα extend to functions of rectangularcoordinates on �δ
t∗ belonging to some Sobolev space H3, then the standardlocal existence theorem (which is stated and proved in rectangular coordinates)applies and we obtain an extension of the solution to a development containingan extension of all null hypersurface Cu for u ∈ [0, δ], up to a value t1 of tfor some t1 > t∗, which contradicts with the definition of t∗ and thereforet∗ = s∗. This completes the proof of Theorem 3.1 and the main theorem ofthe paper.
123
On the formation of shocks for quasilinear wave equations
Acknowledgments The authors are grateful to three anonymous referees, who carefully reada previous version of this paper and suggested many valuable improvements and corrections.S. Miao is supported by NSF Grant DMS-1253149 to The University of Michigan. P. Yu issupported by NSFC 11522111 and NSFC 11271219. The research of S. Miao was in its initialphase supported by ERC Advanced Grant 246574 “Partial Differential Equations of ClassicalPhysics”.
References
1. Alinhac, S.: Temps de vie des solutions régulières des équations d’Euler compressiblesaxisymétriques en dimension deux. Invent. Math. 111(3), 627–670 (1993)
2. Alinhac, S.: Blowup of small data solutions for a quasilinear wave equation in two spacedimensions. Ann. Math. (2) 149(1), 97–127 (1999)
3. Alinhac, S.: Blowup of small data solutions for a class of quasilinear wave equations intwo space dimensions. II. Acta Math. 182(1), 1–23 (1999)
4. Christodoulou, D.: The Action Principle and Partial Differential Equations. Annals ofMathematics Studies, vol. 146. Princeton University Press, Princeton, NJ (2000)
5. Christodoulou, D.: The Formation of Shocks in 3-Dimensional Fluids. Monographs inMathematics. European Mathematical Society, Zürich (2007)
6. Christodoulou, D.: The Formation of Black Holes in General Relativity. Monographs inMathematics. European Mathematical Society, Zürich (2009)
7. Christodoulou, D., Klainerman, S.: The Global Nonlinear Stability of Minkowski Space.Princeton Mathematical Series, vol. 41. Princeton University Press, Princeton, NJ (1993)
8. Christodoulou, D., Miao, S.: Compressible Flow and Euler’s Equations. (Monograph, 602pp.). Surveys in Modern Mathematics, vol. 9. International Press, Somerville, MA; HigherEducation Press, Beijing (2014)
9. Holzegel, G., Klainerman, S., Speck, J., Wong, W.: Shock Formation in Small-Data Solu-tions to 3D Quasilinear Wave Equations: An Overview. J. Hyperbolic Differ. Equ. 13(1),1–105 (2016)
10. John, F.: Blow-up of radial solutions of utt = c2(ut )�u in three space dimensions. Mat.Apl. Comput. 4(1), 3–18 (1985)
11. John, F.: NonlinearWave Equations, Formation of Singularities. University Lecture Series,vol. 2. AMS, Providence (1990)
12. Klainerman, S.: Global existence for nonlinearwave equations. Commun. PureAppl.Math.33(1), 43–101 (1980)
13. Klainerman, S., Rodnianski, I.: On the formation of trapped surfaces. Acta Math. 208(2),211–333 (2012)
14. Speck, J.: Shock formation in small-data solutions to 3D quasilinear wave equations.arXiv:1407.6320v1 (preprint)
![QUASILINEAR AND HESSIAN EQUATIONS … · arXiv:math/0501483v1 [math.AP] 26 Jan 2005 QUASILINEAR AND HESSIAN EQUATIONS OF LANE–EMDEN TYPE NGUYEN CONG PHUC AND IGOR E. VERBITSKY∗](https://static.documents.pub/doc/80x56/5c3061d709d3f25b438b6b3a/quasilinear-and-hessian-equations-arxivmath0501483v1-mathap-26-jan-2005.jpg)
